Topics on Concentration Phenomena and Problems with Multiple Scales

Lecture Notes of the Unione Matematica Italiana 2 Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Mat...

Author: Andrea Braides | Valeria Chiadò Piat

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Lecture Notes of the Unione Matematica Italiana

2

Editorial Board Franco Brezzi (Editor in Chief) Dipartimento di Matematica Università di Pavia Via Ferrata 1 27100 Pavia, Italy e-mail: [email protected]

Persi Diaconis Department of Statistics Stanford University Stanford, CA 94305-4065, USA e-mail: [email protected], [email protected]

John M. Ball Mathematical Institute 24-29 St Giles' Oxford OX1 3LB United Kingdom e-mail: [email protected]

Nicola Fusco Dipartimento di Matematica e Applicazioni Università di Napoli "Federico II", via Cintia Complesso Universitario di Monte S.Angelo 80126 Napoli, Italy e-mail [email protected]

Alberto Bressan Department of Mathematics Penn State University University Park State College PA. 16802, USA e-mail: [email protected]

Carlos E. Kenig Department of Mathematics University of Chicago 1118 E 58th Street, University Avenue Chicago IL 60637, USA e-mail: [email protected]

Fabrizio Catanese Mathematisches Institut Universitätstraße 30 95447 Bayreuth, Germany e-mail: [email protected]

Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa, Italy e-mail: [email protected]

Carlo Cercignani Dipartimento di Matematica Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano, Italy e-mail: [email protected]

Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam, The Netherlands e-mail: [email protected]

Corrado De Concini Dipartimento di Matematica Università di Roma "La Sapienza" Piazzale Aldo Moro 2 00133 Roma, Italy e-mail: [email protected]

Cédric Villani Ecole Normale Supérieure de Lyon 46, allée d'Italie 69364 Lyon Cedex 07 France e-mail: [email protected]

The Editorial Policy can be found at the back of the volume.

Andrea Braides . Valeria Chiadò Piat (Eds.)

Topics on Concentration Phenomena and Problems with Multiple Scales

ABC

Editors Andrea Braides Dipartimento di Matematica Università di Roma "Tor Vergata" via della Ricerca Scientifica 1 00133 Roma Italy e-mail: [email protected]

Valeria Chiadò Piat Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italy e-mail: [email protected]

Library of Congress Control Number: 2006928616 Mathematics Subject Classification (2000): 49-06, 49J40, 49J45, 49K 20, 49J55, 28A 80, 74Q 05, 35B 27, 35B 25, 35B 40 ISSN print edition: 1862-9113 ISSN electronic edition: 1862-9121 ISBN-10 3-540-36241-x Springer Berlin Heidelberg New York ISBN-13 978-3-540-36241-8 Springer Berlin Heidelberg New York

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Preface

The research group ‘Homogenization Techniques and Asymptotic Methods for Problems with Multiple Scales’, co-ordinated by Valeria Chiad` o Piat and funded by INdAM-GNAMPA (Istituto Nazionale di Alta Matematica-Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni), operated from 2001 to 2005, involving in its activities a number of young Italian mathematicians, mainly interested in problems in the Calculus of Variations and Partial Differential Equations. One of the initiatives of that group has been the organization of a number of schools. Those in the years 2001-2003, whose lecture notes are gathered in this book, had been devoted to problems with oscillations and concentrations, while the schools in the years 2004-2005 covered a range of topics of Applied Mathematics. The first school in Turin, 17–21 September 2001, bearing the name of the research group and devoted to problems with multiple scales, was partially disrupted by the events of September 11 of that year, one speaker, Gilles Francfort, finding himself grounded in Los Angeles, and two other speakers, Andrey Piatnitski and Gregory Chechkin, slowed down in their car trip to Italy by the tightening of the borders around the European Community. The two remaining speakers managed however to enlarge their courses to cover some extra material, encouraged by the receptive audience. The course of Andrea Braides was devoted to the description of the behaviour of variational problems on lattices as the lattice spacing tends to zero, and the various multiscale behaviours that may be obtained from this process; that of Anneliese Defranceschi to energies with competing bulk and surface interactions. The extra lectures are not included in these notes, but some of them constitute part of the material in the book ‘Γ -convergence for Beginners’ (Oxford U.P., 2002) by Braides. The course of Francfort, on H-measures, was later recovered in a ‘Part II’ of the School held at IAC in Rome, December 3–5, 2001, together with a contribution of Roberto Peirone on homogenization on fractals. Here we also include the text of the two courses of Piatnitski and Chechkin, while the lecture notes of the course by Francfort have appeared as a chapter of the book ‘Variational Problems in Materials Science’, Birkh¨ auser, 2006.

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Preface

The second part of the present notes covers the content of the subsequent school on ‘Concentration Phenomena for Variational Problems’ held at the Department of Mathematics of the University of Rome ‘La Sapienza’, September 1–5, 2003 (co-organized by A. Braides, which explains why he appears both as an editor and as a contributor). Scope of the School was to present different problems in the Calculus of Variations depending on a small parameter ε, that exhibit a dramatic ‘change of type’ as this parameter tends to 0, that is best described by the ‘concentration’ of some quantity at some lowerdimensional set. The courses of Sylvia Serfaty and Didier Smets treat the case of Ginzburg-Landau energies. In a two-dimensional setting it is known that the concentration of Jacobians of minimizers at points can be interpreted as the arising of ‘vortices’. A novel method envisaged by Sandier and Serfaty shows how the limit motion of these vortices can be described by making use of Γ -convergence. On the other hand, Smets’s course focuses on the information that can be obtained by looking at the fine behaviour of solutions of the Allen-Cahn equations, and concerns the motion in any dimension. The use of Γ -convergence as a way to describe the concentration of maximizers of problems with sub-critical growth is also the subject of the third course by Adriana Garroni. Here the concentrating quantity is not a Jacobian (the problem is scalar), but a suitable scaling of the square of the gradient of the maximizer, that converges as a measure to a sum of Dirac masses. This phenomenon has been previously described by means of the Concentration-Compactness alternative, and this ‘version’ by Γ -convergence gives a new interpretation of the results. The course of Sylvia Serfaty, originally programmed for this school, had to be postponed to a subsequent spin-off ‘School on Geometric Evolution Problems’ held at the Department of Mathematics of the University of Rome ‘Tor Vergata’, January 26–28, 2004 (with the same organizing team, and an additional course by Giovanni Bellettini) but is considered essentially part of the September 2003 School, and that is why it is included here. Other two courses, whose notes are not presented here, were held at the School by Giovanni Alberti and Halil Mete Soner. The course of Soner followed the notes of a previous school and can be found in his Lecture Notes ‘Variational and dynamic problems for the Ginzburg-Landau functional. Mathematical aspects of evolving interfaces’ (Lecture Notes in Math. 1812, Springer, 2003, 177– 233). Alberti’s presentation is partly covered by his review paper ‘A variational convergence result for Ginzburg-Landau functionals in any dimension’ (Boll. Un. Mat Ital. 4 (2001), 289–310). As a final acknowledgement, it must be mentioned that these schools had been additionally jointly sponsored by the Rome and Milan Units of the National Project ‘Calculus of Variations’.

Rome and Turin, February 2006

Andrea Braides Valeria Chiad` o Piat

Contents

Part I Problems with multiple scales From discrete systems to continuous variational problems: an introduction Andrea Braides and Maria Stella Gelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Discrete problems with limit energies defined on Sobolev spaces . . . . 1.1 Discrete functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Equivalent energies on Sobolev functions . . . . . . . . . . . . . . . . . . . . 1.3 Convex energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Non-convex energies with superlinear growth . . . . . . . . . . . . . . . . 1.5 A general convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Convergence of minimum problems . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Energies depending on second-difference quotients . . . . . . . . . . . 2 Limit energies on discontinuous functions: two examples . . . . . . . . . . . 2.1 The Blake-Zisserman model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Second-difference interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Lennard-Jones potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Renormalization-group approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 General convergence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nearest-neighbour interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Long-range interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relaxation for bulk and interfacial energies Anneliese Defranceschi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Lower semicontinuity - The direct method of the Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 6 6 7 8 11 19 21 28 35 36 36 51 52 58 58 58 60 73 75 79 79 81 83

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Preliminaries. Functions of bounded variation and special functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Measure spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Sets of finite perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 Structure of BV functions. Approximate discontinuity points and approximate jump points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.6 Special functions of bounded variation . . . . . . . . . . . . . . . . . . . . . . 92 4.7 BV (SBV ) functions in one dimension . . . . . . . . . . . . . . . . . . . . . 92 5 Lower semicontinuity and relaxation in one dimension . . . . . . . . . . . . . 93 5.1 A characterization of l.s.c. for functionals defined on SBV (I). Relaxation on BV (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2 A sufficient condition for l.s.c. for functionals defined on SBV (I). Relaxation in BV (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6 Lower semicontinuity and relaxation in higher dimension . . . . . . . . . . 106 7 Lower semicontinuity and relaxation in higher dimension and for vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Upper bound for the relaxed functional . . . . . . . . . . . . . . . . . . . . . 119 7.2 De Giorgi-Letta criterion to prove measure properties of the relaxed functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3 Integral representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.4 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 8 Relaxation in higher dimension (isotropy assumptions) . . . . . . . . . . . . 129 9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 9.1 Appendix A: Lower semicontinuous envelope . . . . . . . . . . . . . . . . 132 9.2 Appendix B: Proof of (73) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.3 Appendix C: Proof of Step 3 in Proposition 67 . . . . . . . . . . . . . . 135 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Convergence of Dirichlet forms on fractals Roberto Peirone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2 Construction of an energy on the Gasket . . . . . . . . . . . . . . . . . . . . . . . . 141 3 Dirichlet forms on finitely ramified fractals . . . . . . . . . . . . . . . . . . . . . . 152 4 Main properties of renormalization and harmonic extension . . . . . . . . 164 5 Homogenization on the Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6 Homogenization on general fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Homogenization in perforated domains Gregory A. Chechkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 1 Appearence of a “term etrange”. Dirichlet problems in domains with low concentration of holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 1.1 Basic notation and setting of the problem . . . . . . . . . . . . . . . . . . . 190

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1.2 The homogenization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 Homogenization problems in perforated domain with oscillating Fourier boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 2.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 2.2 Formal asymptotic analysis of the problem . . . . . . . . . . . . . . . . . . 194 2.3 Main estimates and results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 2.4 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 2.5 Justification of asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 2

Homogenization of random non stationary parabolic operators Andrey Piatnitski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 2 Factorization of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3 A priori estimates for the factorized equation . . . . . . . . . . . . . . . . . . . . 214 4 Auxiliary problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 5 Homogenization of the factorized equation . . . . . . . . . . . . . . . . . . . . . . . 218 6 Homogenization of the original equation . . . . . . . . . . . . . . . . . . . . . . . . . 223 7 Equations with diffusion driving process . . . . . . . . . . . . . . . . . . . . . . . . . 225 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Part II Problems with concentration Γ -convergence for concentration problems Adriana Garroni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 2 Two classical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 2.1 Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 2.2 Capacity and isoperimetric inequality for the capacity . . . . . . . . 238 3 The general problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 3.2 Generalized Sobolev inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.3 Concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.4 Γ + -convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.5 The concentration result in terms of Γ + -convergence . . . . . . . . . 246 4 Identification of the concentration point . . . . . . . . . . . . . . . . . . . . . . . . . 252 4.1 The Green’s function and the Robin function . . . . . . . . . . . . . . . . 253 4.2 Asymptotic expansion in Γ + -convergence . . . . . . . . . . . . . . . . . . . 256 4.3 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4.4 Harmonic transplantation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5 Irregular domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

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Gamma-convergence of gradient flows and applications to Ginzburg-Landau vortex dynamics Sylvia Serfaty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 1.1 Presentation of the Ginzburg-Landau model . . . . . . . . . . . . . . . . . 267 1.2 Gamma-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 1.3 Γ -convergence of Ginzburg-Landau . . . . . . . . . . . . . . . . . . . . . . . . 271 2 The abstract result for Γ -convergence of gradient-flows . . . . . . . . . . . . 275 2.1 The abstract situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 2.2 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 2.3 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 2.4 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 2.5 Application to Ginzburg-Landau . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 3 Proof of the additional conditions for Ginzburg-Landau . . . . . . . . . . . 284 3.1 A product-estimate for Ginzburg-Landau . . . . . . . . . . . . . . . . . . . 284 3.2 Idea of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 3.3 Application to the dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 3.4 Proof of the construction 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 4 Extensions of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.1 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.2 Second order questions - stability issues . . . . . . . . . . . . . . . . . . . . 290 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 PDE analysis of concentrating energies for the GinzburgLandau equation Didier Smets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 2 The monotonicity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 2.1 Scaling properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 2.2 The monotonicity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 2.3 Consequences for µ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 3 Regularity theorems for (GL)ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 3.1 Bochner, small energy regularity and clearing-out . . . . . . . . . . . . 299 3.2 Consequences for µ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 4 Potential and modulus estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 5 Σµ is a minimal surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 5.1 Classical and weak notions of mean curvature . . . . . . . . . . . . . . . 311 5.2 Deriving the curvature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

List of Contributors

Andrea Braides Dipartimento di Matematica, Universit` a di Roma “Tor Vergata”, via della ricerca scientifica, 1, 00133 Roma, Italy [email protected] Gregory A. Chechkin Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119992, Russia [email protected] Anneliese Defranceschi Dipartimento di Matematica, Universit` a di Trento, Via Sommarive, 14, Povo, Trento, Italy [email protected] Adriana Garroni Dipartimento di Matematica, Universit` a di Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy [email protected] Maria Stella Gelli Dipartimento di Matematica “L. Tonelli”, Universit` a di Pisa, Largo B. Pontecorvo, 5, Pisa, Italy [email protected]

Roberto Peirone Dipartimento di Matematica, Universit` a di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133, Roma, Italy [email protected] Andrey Piatnitski Narvik University College, Postboks 385, 8505 Narvik, Norway and P.N. Lebedev Physical Institute of RAS, Leninski pr., 53, Moscow 119991, Russia [email protected] Sylvia Serfaty Courant Institute of Mathematical Sciences, 251 Mercer St, New York NY 10012, USA [email protected], Didier Smets Laboratoire Jacques-Louis Lions, Université de Paris 6, 4 place Jussieu, Boite Courier 187, 75252 Paris, France [email protected]

Part I

Problems with multiple scales

From discrete systems to continuous variational problems: an introduction Andrea Braides1 and Maria Stella Gelli2 1

2

Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” via della ricerca scientifica, 1, 00133 Roma, Italy e-mail: [email protected], http://www.mat.uniroma2.it/ ˜braides Dipartimento di Matematica “L. Tonelli”, Universit` a di Pisa Largo B. Pontecorvo, 5, Pisa, Italy e-mail: [email protected]

Introduction These lecture notes cover the content of a course given by the first author at the School Homogenization Techniques and Asymptotic Methods for Problems with Multiple Scales in Turin, 17–21 September 2001, and previously delivered at SISSA, Trieste, jointly by both authors in a slightly enlarged version. In them, we treat the problem of the description of variational limits of discrete problems in a one-dimensional setting. Even though this is a simplified setting, the results that we are going to illustrate contain many of the features that we may encounter in a higher-dimensional framework. After this course had been held a number of papers have appeared, among which a general compactness theorem for variational limits of discrete systems in any dimension by Alicandro and Cicalese [3], accurate estimates on pointwise limits for classes of lattice systems by Blanc, Le Bris and Lions and applications to atomistic computations [10, 11, 34], an interesting analysis of the Cauchy-Born hypothesis by Friesecke and Theil [32], applications to higher-dimensional homogenization of networks by Braides and Francfort [19], to percolation problems by Braides and Piatnitski [25], to anti-phase boundaries in spin systems [1], thin discrete objects [2], etc. Not only those applications have not made the present notes obsolete, but on the contrary they have more motivated them as a necessary introduction to understand the non-trivial effects of the ‘microscopic’ dimension. Given n ∈ N we consider energies of the general form En ({ui }) =

n n−j j=1 i=0

λn ψnj

u

− ui jλn

i+j

4

Andrea Braides and Maria Stella Gelli

defined on (n+1)-tuples {ui }. We may view {ui } as a discrete function defined on a lattice covering a fixed interval [0, L] by introducing points xni = iλn (λn = L/n is the lattice spacing). If we picture the set {xni } as the reference configuration of an array of material points interacting through some forces, and let ui represent the displacement of the i-th point, then ψnj can be thought as the energy density of the interaction of points with distance jλn (j lattice spacings) in the reference lattice. Note that the only assumption we make is that ψnj depends on {ui } through the differences ui+j −ui , but we find it more convenient to highlight its dependence on the ‘discrete difference quotients’ ui+j − ui . jλn One must not be distracted from this notation and should note the generality of the approach. Our goal is to describe the behaviour of problems of the form n ui fi : u0 = U0 , un = UL min En ({ui }) − i=0

(and similar), and to show that for a quite general class of energies these problems have a limit continuous counterpart. Here {fi } represents the external forces and U0 , UL are the boundary conditions at the endpoints of the interval (0, L). More general statements and different problems can be also obtained. To make this asymptotic analysis precise, we use the notation and methods of Γ -convergence, for which we refer to the book by A. Braides Γ -Convergence for Beginners (a more complete theoretical introduction can be found in the book by G. Dal Maso An Introduction to Γ -convergence). We will show that, upon suitably identifying discrete functions {ui } with suitable (possibly discontinuous) interpolations, the free energies En ‘Γ -converge’ to a limit energy F . As a consequence we obtain that minimizers of the problem above are ‘very close’ to minimizers of L f u dt : u(0) = U0 , u(L) = UL . min F (u) − 0

The energies F can be explicitly identified by a series of operations on the functions ψnj . In order to give an idea of how F can be described, we first consider the case when only nearest-neighbour interactions are taken into account: n−1 u i+1 − ui . λ n ψn En ({ui }) = λn i=0 In this case, the limit functional F can be described by introducing for each n a ‘threshold’ Tn such that Tn → +∞ and λn Tn → 0, and defining a limit bulk energy density

From discrete systems to continuous energies

5

f (z) = lim (convex envelope of ψñ (z)), n

and a limit interfacial energy density ˜ g(z) = lim (subadditive envelope of λn ψñ n

where

ψñ (z) =

ψn (z) +∞

z ), λn

if |z| ≤ Tn otherwise,

ψn (z) +∞

˜ ψñ (z) =

if |z| ≥ Tn otherwise.

Note the crucial separation of scales argument: essentially, the limit behaviour of ψn (z) defines the bulk energy density, while λn ψn (z/λn ) determines the interfacial energy. The limit F is defined (up to passing to its lower semicontinuity envelope) on piecewise-Sobolev functions as f (u ) dt + g(u(t+) − u(t−)), F (u) = (0,L)

S(u)

where S(u) denotes the set of discontinuity points of u. Hence, we have a limit energy with two competing contributions of a bulk part and of an interfacial energy. In this form we can recover fracture and softening phenomena. The description of the limit energy gets more complex when not only nearest-neighbour interactions come into play. We first examine the case when interactions up to a fixed order K are taken into account: En ({ui }) =

K n−j

λn ψnj

u

− ui jλn

i+j

j=1 i=0

(or, equivalently, ψnj = 0 if j > K). The main idea is to show that (upon some controllable errors) we can find a lattice spacing ηn (possibly much larger than λn ) such that En is ‘equivalent’ (as Γ -convergence is concerned) to a nearest-neighbour interaction energy on a lattice of step size ηn , of the form E n ({ui }) =

m−1 i=0

ηn ψ n

u

i+1

− ui

ηn

,

and to which then the recipe above can be applied. The crucial points here are the computation of ψ n and the choice of the scaling ηn . In the case of next-to-nearest neighbours this computation is particularly simple, as it consists in choosing ηn = 2λn and in ‘integrating out the contribution of first neighbours’: in formula, ψ n (z) = ψn2 (z) +

1 min{ψn1 (z1 ) + ψn1 (z2 ) : z1 + z2 = 2z}. 2

6


In a sense this is a formula of relaxation type. If K > 2 then the formula giving ψn resembles more a homogenization formula, and we have to choose ηn = Kn λn with Kn large. In this case the reasoning that leads from En to E n is that the overall behaviour of a system of interacting points will behave as clusters of large arrays of neighbouring points interacting through their ‘extremities’. When the number of interaction orders we consider is not bounded the description becomes more complex. In particular, additional non-local terms may appear in F . Note that first order Γ -limits may not capture completely the behaviour of minimizers for variational problems as above. Additional information, as phase transitions, boundary layer effects and multiple cracking, may be extracted from the study of higher-order Γ -limits.

1 Discrete problems with limit energies defined on Sobolev spaces 1.1 Discrete functionals We will consider the limit of energies defined on one-dimensional discrete systems of n points as n tends to +∞. In order to define a limit energy on a continuum we parameterize these points as a subset of a single interval (0, L). Set L i xni = L = iλn , i = 0, 1, . . . , n. (1) λn = , n n We denote In = {xn0 , . . . , xnn } and by An (0, L) the set of functions u : In → R. If n is fixed and u ∈ An (0, L) we equivalently denote ui = u(xni ). Given K ∈ N with 1 ≤ K ≤ n and functions f j : R → [0, +∞], with j = 1, . . . , K, we will consider the related functional E : An (0, L) → [0, +∞] given by K n−j f j (ui+j − ui ). (2) E(u) = j=1 i=0

Note that E can be viewed simply as a function E : Rn → [0, +∞]. An interpretation with a physical flavour of the energy E is as the internal interaction energy of a chain of n+1 material points each one interacting with its K-nearest neighbours, under the assumption that the interaction energy densities depend only on the order j of the interaction and on the distance between the two points ui+j − ui in the reference configuration. If K = 1 then each point interacts with its nearest neighbour only, while if K = n then each pair of points interacts.


7

Remark 1. From elementary calculus we have that E is lower semicontinuous if each f j is lower semicontinuous, and that E is coercive on bounded sets of An (0, L). 1.2 Equivalent energies on Sobolev functions We will describe the limit as n → +∞ of sequences (En ) with En : An (0, L) → [0, +∞] of the general form En (u) =

Kn n−j

fnj (ui+j − ui ).

(3)

j=1 i=0

Since each functional En is defined on a different space, the first step is to identify each An (0, L) with a subspace of a common space of functions defined on (0, L). In order to identify each discrete function with a continuous counterpart, we extend u by u ˜ : (0, L) → R as the piecewise-affine function defined by u ˜(s) = ui−1 +

ui − ui−1 (s − xi−1 ) λn

if s ∈ (xi−1 , xi ).

(4)

In this case, An (0, L) is identified with those continuous u ∈ W 1,1 (0, L) (actually, in W 1,∞ (0, L)) such that u is affine on each interval (xi−1 , xi ). Note moreover that we have ui − ui−1 (5) u ˜ = λn ñ . on (xi−1 , xi ). If no confusion is possible, we will simply write u in place of u It is convenient to rewrite the dependence of the energy densities in (3) with respect to difference quotients rather than the differences ui+j − ui . We then write Kn n−j u i+j − ui , (6) λn ψnj En (u) = jλn j=1 i=0 where ψnj (z) =

1 j f (jλn z). λn n

With the identification of u with u ˜, En may be viewed as an integral functional defined on W 1,1 (0, L). In fact, for fixed j ∈ {0, . . . , K − 1}, k ∈ {0, . . . , n − 1} and i such that i ≤ k < i + j we have 1 ui+j − ui = jλn j

i−k+j−1 m=i−k

1 uk+m+1 − uk+m = λn j

for all x ∈ (xnk , xnk+1 ), so that

i−k+j−1 m=i−k

u ˜ (x + mλn )

8


λn ψ

j

u

i+j−1 xn 1 i−k+j−1 k+1 − ui 1 = ψj u ˜ (x + mλn ) dx. jλn j j xn k

i+j

k=i

m=i−k

We then get n−j

λn ψnj

i=0

u

j−1 L−(j−1−l)λn

− ui 1 = jλn j

i+j

ψnj lλn

l=0

1 j−1−l j

u ˜ (x + kλn ) dx.

k=−l

and the equality u), En (u) = Fn (˜

(7)

where ⎧ j−1 j−1−l Kn ⎪ 1 L−(j−1−l)λn j 1 ⎪ ⎪ ψn v (x + kλn ) dx ⎪ ⎨ j lλn j k=−l Fn (v) = j=1 l=0 ⎪ ⎪ if v ∈ An (0, L) ⎪ ⎪ ⎩ +∞ otherwise. (8) Note that in the particular case Kn = 1 we have (set ψn = ψn1 ) ⎧ ⎨ Fn (v) =

L

ψn (v )dx

⎩ +∞

if v ∈ An (0, L)

0

(9)

otherwise.

Definition 1 (Convergence of discrete functions and energies). With the identifications above we will say that un converges to u (respectively, ñ converge to u (respectively, in L1 , in in L1 , in measure, in W 1,1 , etc.) if u measure, weakly in W 1,1 , etc.), and we will say that En Γ -converges to F (respectively, with respect to the convergence in L1 , in measure, weakly in W 1,1 , etc.) if Fn Γ -converges to F (respectively, with respect to the convergence in L1 , in measure, weakly in W 1,1 , etc.). We refer to [14] for the notation and main definitions of Γ -convergence.

1.3 Convex energies We first treat the case when the energies ψnj are convex. We will see that in the case of nearest neighbours, the limit is obtained by simply replacing sums by integrals, while in the case of long-range interactions a superposition principle holds. For simplicity we suppose that the energy densities do not depend on n; i.e., ψnj = ψ j .


9

Nearest-neighbour interactions We start by considering the case K = 1, so that the functionals En are given by n−1 u i+1 − ui . (10) λn ψ En (u) = λn i=0 The integral counterpart of En is given by ⎧ L ⎨ ψ(v )dx if v ∈ An (0, L) Fn (v) = 0 ⎩ +∞ otherwise.

(11)

Note that Fn depends on n only through its domain An (0, L). The following result states that as n approaches ∞ the identification of En with its continuous analogue is complete. Theorem 1. Let ψ : R → [0, +∞) be convex and let En be given by (10). (i) The Γ -limit of En with respect to the weak convergence in W 1,1 (0, L) is given by F defined by ψ(u ) dx. (12) F (u) = (0,L)

(ii) If lim

|z|→∞

ψ(z) = +∞ |z|

(13)

then the Γ -limit of En with respect to the convergence in L1 (0, L) is given by F defined by ⎧ ⎨ ψ(u ) dx if u ∈ W 1,1 (0, L) (14) F (u) = (0,L) ⎩ +∞ otherwise on L1 (0, L). Proof. (i) The functional F defines a weakly lower semicontinuous functional on W 1,1 (0, L) and clearly Fn ≥ F ; hence also we have Γ -lim inf j Fj (u) ≥ F (u). Conversely, fixed u ∈ W 1,1 (0, L) let un ∈ An (0, L) be such that un (xni ) = u(xni ). By convexity we have

xn i+1

xn i

1 xni+1 u(xn ) − u(xn ) i+1 i ; ψ(u ) dt ≥ λn ψ u dt = λn ψ λn xni λn

hence, summing up,

0

L

ψ(u ) dt ≥ En (un ).

10


This shows that (un ) is a recovery sequence for F . (ii) If (13) holds then the sequence (En ) is equi-coercive on bounded sets of L1 (0, L) with respect to the weak convergence in W 1,1 (0, L), from which the thesis is easily deduced. Long-range interactions Let now K ∈ N be fixed. The energies En take the form En (u) =

K n−j

λn ψnj

u

− ui . jλn

i+j

j=1 i=0

(15)

Theorem 2. Let ψ j : R → [0, +∞) be convex and let En be given by (15). Let ψ 1 satisfy ψ 1 (z) = +∞ (16) lim |z|→∞ |z| then the Γ -limit of En with respect to the convergence in L1 (0, L) is given by F defined by ⎧ ⎨ ψ(u ) dx if u ∈ W 1,1 (0, L) (17) F (u) = (0,L) ⎩ +∞ otherwise on L1 (0, L), where ψ=

K

ψj .

(18)

j=1

Proof. Note that (En ) is equi-coercive on bounded set of L1 (0, L) as in the proof of Theorem 1. Then it suffices to check the Γ -limit on W 1,1 (0, L). To prove the Γ -liminf inequality let un u weakly in W 1,1 (0, L). Then, for every j ∈ {0, . . . , K} and l ∈ {0, . . . , j − 1}, also the convex combination j−1−l 1 u ñ (x + kλn ) j

uj,l n =

k=−l

1,1 converge weakly to u in Wloc (0, L). By (7) then we have, for all fixed η > 0,

lim inf En (u) ≥ lim inf n

j−1 K 1

n

≥

j=1

j−1 K 1 j=1 l=0

≥

j−1 K j=1 l=0

j 1 j

j

l=0

L−η

η

L−η

lim inf n

dx ψnj (uj,l ) n ψnj (uj,l n ) dx

η

L−η j

ψ (u )dx = η

η

L−η

ψ(u ) dt.


11

The liminf inequality follows by the arbitrariness of η. Again, fixed u ∈ W 1,1 (0, L) let un ∈ An (0, L) be such that un (xni ) = u(xni ). By Jensen’s inequality, K n−j 1 xni+j 1 xni+j λn ψ u dt ≤ ψ j (u )dt En (un ) = n n jλ j n x x i i j=1 i=0 j=1 i=0

=

K n−j

j

n−j K 1

xn i+j

j=1

j

i=0

xn i

ψ (u )dt ≤ j

K j=1

L

ψ j (u )dt,

0

which implies the limsup inequality. 1.4 Non-convex energies with superlinear growth We now investigate the effects of the lack of convexity, always in the framework of limits defined on Sobolev spaces. Again we suppose that the energy densities do not depend on n; i.e., ψnj = ψ j , but are not necessarily convex. Nearest-neighbour interactions We consider the case K = 1. In this case the only effect of the passage from the discrete setting to the continuum is a convexification of the integrand. Theorem 3. Let ψ : R → [0, +∞) be a Borel function satisfying (13). Let En be given by (10); then the Γ -limit of En with respect to the convergence in L1 (0, L) is given by F defined by ⎧ ⎨ ψ ∗∗ (u ) dx if u ∈ W 1,1 (0, L) F (u) = (19) (0,L) ⎩ +∞ otherwise on L1 (0, L). Proof. The Γ -liminf inequality immediately follows as in the proof of Theorem 1(i). As for the limsup inequality, first note that if u ∈ W 1,1 (0, L) and ψ(u ) = ∗∗ ψ (u ) a.e. then we may simply take un as in the proof of Theorem 1(i), so that for such u we have Γ -limn En (u) = F (u). If ψ is lower semicontinuous and u is affine with u = z, let z1 , z2 ∈ R and λ ∈ [0, 1] be such that z = λz1 + (1 − λ)z2 , and

ψ(z1 ) = ψ ∗∗ (z1 ), ψ(z2 ) = ψ ∗∗ (z2 )

12


ψ ∗∗ (z) = λψ(z1 ) + (1 − λ)ψ(z2 ). Then there exists uj weakly converging to u such that uj ∈ {z1 , z2 } and F (u) = limj F (uj ). By the lower semicontinuity of the Γ -limsup we then have Γ - lim sup En (u) ≤ lim inf Γ - lim sup En (uj ) = lim inf F (uj ) = F (u), j

n

j

n

as desired. If ψ is not lower semicontinuous then suitable z1,j and z2,j must be chosen such that uj weakly converges to u, uj ∈ {z1,j , z2,j } and F (u) = limj F (uj ). To conclude the proof it remains to suitably approximate any function u ∈ W 1,1 (0, L) by some its affine interpolations (uk ) and remark that by the convexity of F we have F (u) = limk F (uk ). Next-to-nearest neighbour interactions In the non-convex setting, the case K = 2 offers an interesting way of describing the two-level interactions between first and second neighbours. Such description is more difficult in the case K ≥ 3. Essentially, the way the limit continuum theory is obtained is by first integrating-out the contribution due to nearest neighbours by means of an inf-convolution procedure and then by applying the previous results to the resulting functional. Theorem 4. Let ψ 1 , ψ 2 : R → [0, +∞) be Borel functions such that ψ 1 (z) = +∞, |z|→∞ |z| lim

(20)

and let En (u) : An (0, L) → [0, +∞) be given by En (u) =

n−1

λn ψ 1

u

i+1

i=0

− ui

λn

+

n−2 i=0

λn ψ 2

u

− ui 2λn

i+2

(21)

Let ψ˜ : R → [0, +∞) be defined by 1 ˜ ψ(z) = ψ 2 (z) + inf{ψ 1 (z1 ) + ψ 1 (z2 )) : z1 + z2 = 2z} 2 1 2 = inf ψ (z) + (ψ 1 (z1 ) + ψ 1 (z2 )) : z1 + z2 = 2z , 2 and let

ψ = ψ˜∗∗ .

(22) (23)

1

Then the Γ -limit of En with respect to the convergence in L (0, L) is given by F defined by ⎧ ⎨ ψ(u ) dx if u ∈ W 1,1 (0, L) (24) F (u) = (0,L) ⎩ +∞ otherwise on L1 (0, L).


13

Remark 2. (i) The growth conditions on ψ 2 can be weakened, by requiring that ψ 2 : R → R and −c1 ψ 1 ≤ ψ 2 ≤ c2 (1 + ψ 1 ) provided that we still have lim

|z|→∞

ψ(z) = +∞. |z|

(ii) If ψ 1 is convex then ψ˜ = ψ 1 + ψ 2 . If also ψ 2 is convex then we recover a particular case of Theorem 2. Proof. Let u ∈ An (0, L). We have, regrouping the terms in the summation, En (u) =

u 1 u 1 u i+2 − ui i+2 − ui+1 i+2 − ui+1 + ψ1 + ψ1 λn ψ 2 2λn 2 λn 2 λn

n−2 i=0

i even

+

n−2 i=0

u 1 u 1 u i+2 − ui i+2 − ui+1 i+1 − ui + ψ1 + ψ1 λn ψ 2 2λn 2 λn 2 λn

i odd

λn 1 un − un−1 1 1 u1 − u0 ψ + ψ 2 λn 2 λn ⎛ ⎞ n−2 n−2 1⎜ ui+2 − ui ui+2 − ui ⎟ + 2λn ψ˜ 2λn ψ˜ ≥ ⎝ ⎠ 2 2λn 2λn i=0 i=0 +

i even

i odd

⎛ ≥

n−2

1⎜ ⎝ 2

2λn ψ

i=0

i even 2λn [n/2]

=

1 2

0

u

⎞ n−2 ui+2 − ui ⎟ i+2 − ui + 2λn ψ ⎠ 2λn 2λn i=0

ψ(˜ u1 ) dt +

i odd (1+2[n−1/2])λn

ψ(˜ u2 ) dt ,

(25)

λn

where u ˜k , respectively, with k = 1, 2, are the continuous piecewise-affine functions such that ui+2 − ui on (xni , xni+2 ) (26) u ˜k = 2λn for i, respectively, even or odd. Let now un → u in L1 (0, L) and supn En (un ) < +∞; then un u in 1,1 W (0, L). Let uk,n be defined as in (26); as in the proof of Theorem 2, we deduce uk,n u as n → +∞, for k = 1, 2. For every fixed η > 0 by (25) we obtain L−η L−η 1 lim inf ψ(u1,n ) dt + lim inf ψ(u2,n ) dt lim inf En (un ) ≥ n n n 2 η η L−η ≥ ψ(u ) dt, η

14


and the liminf inequality follows by the arbitrariness of η > 0. Now we prove the limsup inequality. By arguing as in the proof of Theorem 3, note that it suffices to treat the case when ψ˜ is lower semicontinu˜ ous, u(x) = zx and ψ(z) = ψ(z). With fixed η > 0 let z1 , z2 be such that z1 + z2 = 2z and 1 ˜ + η. ψ 2 (z) + (ψ 1 (z1 ) + ψ 2 (z2 )) ≤ ψ(z) 2 We define the recovery sequence un as zxni n un (xi ) = z(i − 1)λn + z1 λn

if i is even if i is odd.

We then have En (un ) =

n−1 i=0

≤

λn ψ 1

u (xn ) − u (xn ) n−2 u (xn ) − u (xn ) n i+1 n i n i+2 n i + λn ψ 2 λn 2λ n i=0

L 1 ˜ (ψ (z1 ) + ψ 1 (z2 )) + Lψ 2 (z) ≤ Lψ(z) = Lψ(z) = F (u) 2

as desired. Remark 3 (Multiple-scale effects). The formula defining ψ highlights a doublescale effect. The operation of inf-convolution highlights oscillations on the scale λn , while the convexification of ψ˜ acts at a much larger scale. Long-range interactions We consider now the case of a general K ≥ 1. In this case the effective energy density will be given by a homogenization formula. We suppose for the sake of simplicity that ψ j : R → [0, +∞) are lower semicontinuous and there exists p > 1 such that ψ 1 (z) ≥ c0 (|z|p − 1),

ψ j (z) ≤ cj (1 + |z|p ).

(27)

for all j = 1, . . . , K. Before stating the convergence result we define some energy densities. Let N ∈ N. We define ψN : R → [0, +∞) as follows: −j K N u(i + j) − u(i) 1 ψj ψN (z) = min N j=1 i=0 j

u : {0, . . . , N } → R, u(i) = zi for i ≤ K or i ≥ N − K . (28)

Proposition 1. For all z ∈ R there exists the limit ψ(z) = limN ψN (z).


15

Proof. With fixed z ∈ R, let N, M ∈ N with M > N , and let uN be a minimizer for ψN (z). We define uM : {0, . . . , M } → R as follows: uN (i − lN ) + lN z if lN ≤ i ≤ (l + 1)N (0 ≤ l ≤ M N − 1) uM (i) = zi otherwise. Then we can estimate K M −j 1 j uM (i + j) − uM (i) ψM (z) ≤ ψ M j=1 i=0 j K N −j 1 j uN (i + j) − uN (i) ≤ ψ N j=1 i=0 j

+

K K 1 M − [M/N ]N + k − j j ψ (z) (2K − j)ψ j (z) + N j=1 M j=1

K K 2K N +K ψj (z) + ψj (z) N j=1 M j=1 2K N +K + (1 + |z|p ). ≤ ψN (z) + c N M

≤ ψN (z) +

(29)

Taking first the limsup in M and then the liminf in N we deduce that lim sup ψM (z) ≤ lim inf ψN (z) M

N

as desired. Remark 4. The following properties can be easily proved: (i) c0 (|z|p − 1) ≤ ψ 1 (z) ≤ ψ(z) ≤ c(1 + |z|p ); (ii) ψ is lower semicontinuous; (iii) ψ is convex; (iv) for all N ∈ N we have ψ(z) ≤ ψN (z) + Nc (1 + |z|p ). We can state the convergence theorem. Theorem 5. Let ψ j be as above and let En be defined by (15). Then the Γ limit of En with respect to the convergence in L1 (0, L) is given by F defined by ⎧ ⎨ ψ(u ) dx if u ∈ W 1,p (0, L) (30) F (u) = (0,L) ⎩ +∞ otherwise on L1 (0, L), where ψ is given by Proposition 1.

16


Proof. We begin by establishing the liminf inequality. Let un → u in L1 (0, L) be such that supn En (un ) < +∞. Note that this implies that

L

sup n

|un |p dt < +∞,

0

so that indeed un u weakly in W 1,p (0, L) and hence also un → u in L∞ (0, L). For all k ∈ {0, . . . , N − 1} let |un |p dt. Φn (k) = ((k+N l−2K)λn ,(k+N l+2K)λn )∩(0,L)

l∈N

We have

N −1 k=0

Φn (k) ≤ 2K

L

|un |p dt ≤ c,

0

so that, upon choosing a subsequence if necessary, there exists k such that Φn (k) ≤

c . N

For the sake of notational simplicity we will suppose that this holds with k = 0, and also that n = M N with M ∈ N, so that the inequality above reads M −1 l=0

|un |p dt.

(31)

((N l−2K)λn ,(N l+2K)λn )∩(0,L)

We may always suppose so, upon first reasoning in slightly smaller intervals than (0, L) and then let those intervals invade (0, L). Let vnN be the piecewise-affine function defined on (0, L) such that vnN (0) = un (0) on (xni , xni+1 ), N l + K ≤ i ≤ N (l + 1) − K − 1 (vnN ) = un un ((N l + N − K)λn ) − un ((N l + K)λn ) N (vnN ) = =: zn,l (N − 2K)λn on (N lλn , (N l + K)λn ) ∪ ((N (l + 1) − K)λn , N (l + 1)λn ). The construction of vnN deserves some words of explanation. The function vnN is constructed on each interval (N lλn , N (l+1)λn ) as equal to the function un (up to an additive constant) in the middle interval ((N l+K)λn , (N (l+1)−K)λn ), N in the remaining two intervals. Note and as the affine function of slope zn,l that the construction implies that the function N vn,l : {0, . . . , N } → R

defined by

From discrete systems to continuous energies N vn,l (i) =

17

1 N v ((lN + i)λn ) λn n

N ), and that is a test function for the minimum problem defining ψN (zn,l K N (l+1)−j j=1

=

v N (xn ) − v N (xn ) n n i+j i jλn

λn ψ j

v N ((i + j)) − v N (i) n,l n,l N ≥ N λn ψN (zn,l ). j

i=N l

K N (l+1)−j j=1

λn ψ j

i=N l

(32)

Note moreover that, by H¨ older’s inequality, we have 2K 1−1/p 2K L un L1 (0,L) , |(vnN ) − un | dt ≤ un Lp (0,L) + N N − 2K (0,L) so that, since un (0) = vnN (0) we have a uniform bound vnN − un L∞ (0,L) ≤

C . N

(33)

We have that En (un ) ≥

K N (l+1)−K−j M −1 l=0 j=1

=

=

−

λn ψ j


i=N l+K

K N (l+1)−j M −1 l=0 j=1

un (xn ) − un (xn ) i+j i jλn

i=N l+K

K N (l+1)−K−j M −1 l=0 j=1

λn ψ j

λn ψ j

i=N l

K N M −1 l+K


λn ψ j

l=0 j=1 i=N l

−

K M

N l−j

λn ψ j

l=1 j=1 i=N l−K−j

=:

K N (l+1)−j M −1 l=0 j=1

≥

K M −1

λn ψ j

i=N l

v N (xn ) − v N (xn ) n n i+j i jλn v N (xn ) − v N (xn ) n n i+j i jλn

v N (xn ) − v N (xn ) n n i+j i −In1 − In2 jλn

N N λn ψN (zn,l ) − In1 − In2 ,

l=0 j=1

the last estimate being given by (32).

(34)

18


We give an estimate of the term In1 ; the term In2 can be dealt with similarly. Let i < N l + K ≤ i + j; by the growth conditions on ψ j and the convexity of z → |z|p we have v N (xn ) − v N (xn ) n n i+j i jλn i+j−1 v N (xn ) − v N (xn ) p 1 vnN (xni+1 ) − vnN (xni ) p n i+j n i ≤ c 1+ ≤c 1+ jλn j λn k=i 1 N p ≤ c 1 + K|zn,l | + |un |p dt λn ((N l−2K)λn ,(N l+2K)λn )∩(0,L) ψj

We then deduce by (31) and the fact that |z|p ≤ c(1 + ψN (z)) that In1 ≤

K N M −1 l+K l=0 j=1 i=N l

≤

1 N λn c 1 + ψN (zn,l )+ |un |p dt λn ((N l+K)λn ,(N l+2K)λn )

M −1 c c N + N λn ψN (zn,l ). N N

(35)

l=0

Plugging this estimate and the analogue for In2 into (34) we get M −1 c c N En (un ) ≥ 1 − N λn ψN (zn,l )− . N N

(36)

l=0

By Remark 4(iv) we have ψN (z) ≥ ψ(z) −

c c c (1 + |z|p ) ≥ 1 − ψ(z) − . N N N

From (36) we then have c c N En (un ) ≥ 1 − N λn ψ(zn,l )− . N N

M −1 l=0

Now, note that the piecewise-affine functions uN n defined by uN n (0) = un (0)

and

N (uN n ) = zn,l on (N lλn , N (l + 1)λn )

N are weakly precompact in W 1,p (0, L), so that we may suppose that uN n u . Then by Theorem 1 we have


lim inf n

M −1 l=0

so that

N N λn ψ(zn,l )

= lim inf n

L

ψ((uN n ) ) dt

L

≥

0

ψ((uN ) ) dt,

(37)

0

c lim inf En (un ) ≥ 1 − n N

19

L

ψ((uN ) ) dt −

0

c . N

By (33) and the uniform convergence of un to u we have uN − u L∞ (0,L) ≤

c . N

(38)

By N → +∞ we then obtain the thesis by the lower semicontinuity of letting ψ(u ) dt. To prove the limsup inequality it suffices to deal with the case u(x) = zx since from this construction we easily obtain a recovery sequence for piecewiseaffine functions and then reason by density. To exhibit a recovery sequence for such u it suffices to fix N ∈ N, consider v N a minimum point for the problem defining ψN (z) and define un (xni ) = v N (i − N l)λn + zN lλn We then have

if N l ≤ i ≤ N (l + 1).

K c j ψ (z), lim sup En (un ) ≤ ψN (z) + N j=1 n

and the thesis follows by the arbitrariness of N .

1.5 A general convergence theorem By slightly modifying the proof of Theorem 5 we can easily state a general Γ -convergence result, allowing a dependence also on n for the energy densities. Theorem 6. Let K ≥ 1. Let ψnj : R → [0, +∞) be lower semicontinuous functions and let p > 1 exist such that ψn1 (z) ≥ c0 (|z|p − 1),

ψnj (z) ≤ cj (1 + |z|p ).

(39)

for all j ∈ {1, . . . , K} and n ∈ N. For all N, n ∈ N let ψN,n : R → [0, +∞) be defined by −j K N u(i + j) − u(i) 1 ψnj ψN,n (z) = min N j=1 i=0 j

u : {0, . . . , N } → R, u(i) = zi for i ≤ K or i ≥ N − K . (40)

Suppose that ψ : R → [0, +∞) exists such that

20

Andrea Braides and Maria Stella Gelli ∗∗ ψ(z) = lim lim ψN,n (z) N

n

for all z ∈ R

(41)

(note that this is not restrictive upon passing to a subsequence of n and N ). Let En be defined on An (0, L) by En (u) =

K n−j

λn ψnj

u

− ui . jλn

i+j

j=1 i=0

(42)

Then the Γ -limit of En with respect to the convergence in L1 (0, L) is given by F defined by ⎧ ⎨ ψ(u ) dx if u ∈ W 1,p (0, L) (43) F (u) = (0,L) ⎩ +∞ otherwise on L1 (0, L). Proof. Let un → u in L1 (0, L). We can repeat the proof for the liminf inequality for Theorem 5, substituting ψ j by ψnj and ψN by ψN,n . We then deduce as in (37)–(38) that L c c lim inf lim inf En (un ) ≥ 1 − ψN,n ((uN n ) ) dt − n n N N 0 L c c ψN ((uN ) ) dt − , ≥ 1− N 0 N ∗∗ where ψN = limn ψN,n and the thesis by letting N → +∞. To prove the limsup inequality it suffices to deal with the case u(x) = zx since from this construction we easily obtain a recovery sequence for piecewiseaffine functions and then reason by density. To exhibit a recovery sequence for such u it suffices to fix N ∈ N, consider z1,n , z2,n and ηn ∈ [0, 1] such that ∗∗ (z) = ηn ψN,n (z1,n ) + (1 − ηn )ψN,n (z2,n ), ψN,n

z = ηn z1,n + (1 − ηn )z2,n .

N N and v2,n be minimum points for the problem defining ψN,n (z1,n ) and Let v1,n ψN,n (z2,n ), respectively. For the sake of simplicity assume that there exists m such that mN ηn ∈ N for all n. Define ⎧ N ⎪ ⎨v1,n (i − N l)λn + zmN lλn if mN l ≤ i ≤ mN l + mN ηn n N un (xi ) = v2,n (i − N l − mN ηn )λn + zmN l + z1,n mN ηn λn ⎪ ⎩ if mN l + mN ηn ≤ i ≤ mN (l + 1).

By the growth conditions on ψnj it is easily seen that (zk,n ) are equibounded and that N (i) − zk,n i : i ∈ {0, . . . , N }, n ∈ N} < +∞, sup{vk,n


21

so that un converges to zx uniformly. We then have ∗∗ (z) lim sup En (un ) ≤ L lim sup ψN,n n

n

and the thesis follows by the arbitrariness of N .

1.6 Convergence of minimum problems We first give a general convergence theorem, and subsequently state a finer theorem for next-to-nearest neighbour interactions. Limit minimum problems on the continuum From Theorem 6 we immediately deduce the following theorem. Theorem 7. Let En and F be given by Theorem 6, let f ∈ L1 (0, L) and d > 0. Then the minimum values L f u dt : u(0) = 0, u(L) = d (44) mn = min En (u) + 0

converge to m = min F (u) +

L

f u dt : u(0) = 0, u(L) = d ,

(45)

0

and from each sequence of minimizers of (44) we can extract a subsequence converging to a minimizer of (45). Proof. Since the sequence of functionals (En ) is equi-coercive, it suffices to show that the boundary conditions do not change the form of the Γ -limit; i.e., that for all u ∈ W 1,p (0, L) such that u(0) = 0 and u(L) = d and for all ε > 0 there exists a sequence un such that un (0) = 0, un (L) = d and lim supn En (un ) ≤ F (u) + ε. Let vn → u in L∞ (0, L) be such that limn En (vn ) = F (u). With fixed η > 0 and N ∈ N let Kn ∈ N be such that lim Kn λn = n

For all l ∈ {1, . . . , N } let φN,l n defined by φN,l n (0) = 0, ⎧ ⎪ ⎨1/(Kn λn ) N,l φn = −1/(Kn λn ) ⎪ ⎩ 0

η . N

: [0, L] → [0, 1] be the piecewise-affine function on ((l − 1)Kn λn , lKn λn ) on ((n − lKn )λn , (n − lKn + Kn )λn ) otherwise.

22


Let N,l N,l uN,l n = φn vn + (1 − φn )u.

We have En (uN,l n )

L η+Kλn p ≤ En (un ) + c (1 + |u | ) dt + (1 + |u |p ) dt L−η−Kλn 0 |un |p dt +c ((l−1)Kn −K)λn ,(lKn +K)λn )∩(0,L) + |vn |p dt ((n−lKn −K)λn ,(n−lKn +Kn +K)λn )∩(0,L)

1 |vn − u|p p 0 (Kn λn ) L 2η p ≤ En (un ) + c (1 + |u | ) dt + (1 + |u |p ) dt 0 L−2η +c L

+

|vn |p dt

(((l−2)η/N,((l+1)η/N )∪(L−(l+1)η/N,L−(l−2)η/N ))∩(0,L)

Np +c p vn − u pL∞ (0,L) η for n large enough. Since N l=1

≤2

|un |p dt

((l−2)η/N,((l+1)η/N )∪(L−(l+1)η/N,L−(l−2)η/N )∩(0,L) L

(1 + |vn |p ) dt ≤ c,

0

for all n there exists ln ∈ {1, . . . , N } such that 2η p n En (uN,l ) ≤ E (v ) + c (1 + |u | ) dt + n n n 0

L

(1 + |u |p ) dt

L−2η

Np c + + c p vn − u pL∞ (0,L) N η n Setting un = uN,l we then have n 2η (1 + |u |p ) dt + lim sup En (un ) ≤ F (u) + c

n

0

c (1 + |u |p ) dt + , N L−2η L

and the desired inequality by the arbitrariness of η and N .

Next-to-nearest interactions: phase transitions and boundary layers If the function ψ giving the limit energy density in Theorem 6 is not strictly convex, converging sequences of minimizers of problems of the type (44) may


23

converge to particular minimizers of (45). This happens in the case of nextto-nearest interactions, where the formula giving ψ is of particular help. We examine the case when ψ˜ in (22) is not convex and of minimum problems (44) with f = 0. Upon some change of coordinates it is not restrictive to examine problems of the form mn = min{En (u) : u(0) = 0, u(L) = 0},

(46)

˜ ˜ min ψ˜ = ψ(1) = ψ(−1).

(47)

and to suppose (H1) we have For the sake of simplicity we make the additional assumptions (H2) we have ˜ ψ(z) > 0 if |z| = 1; z1+ , z2+

(48)

z1− , z2−

and such that (H3) there exist unique 1 1 ± ˜ ψ (z1 ) + ψ 1 (z2± ) = min ψ, z1± , z2± = ±2; ψ 2 (±1) + 2 We set M+ = {(z1+ , z2+ ), (z2+ , z1+ )}, M− = {(z1− , z2− ), (z2− , z1− )} M = M+ ∪ M− .

(49) (50)

(H4) we have zi+ = zj− for all i, j ∈ {1, 2}; (H5) all functions are C 1 . Under hypotheses (H1)–(H2) Theorem 6 simply gives that mn → 0 and that the limits u of minimizers satisfy |u | ≤ 1 a.e. We will see that indeed they are ‘extremal’ solutions to the problem min{F (u) : u(0) = 0, u(L) = 0}.

(51)

The effect of the non validity of hypotheses (H3)–(H5) is explained in Remark 6. The key idea is that it is energetically convenient for discrete minimizer ˜ and that every time we have to remain close to the two states minimizing ψ, a transition from one of the two minimal configurations to the other a fixed amount of energy is spent (independent of n). To exactly quantify this fact we introduce some functions and quantities. Definition 2 (Minimal energy configurations). Let z = (z1 , z2 ) ∈ M; we define uz : Z → R by i i z2 + i − z1 , (52) uz (i) = 2 2 and uzn : λn Z → R by

uzn (xni ) = uz (i)λn

(53)

24


Definition 3 (Crease and boundary-layer energies). Let v : Z → R. The right-hand side boundary layer energy of v is B+ (v) = inf min

ψ2

N ∈N

i≥0

u(i + 2) − u(i) 2

+ ψ 1 (u(i + 1) − u(i)) − min ψ˜

: u : N ∪ {0} → R, u(i) = v(i) if i ≥ N ,

The left-hand side boundary layer energy of v is u(i) − u(i − 2) + ψ 1 (u(i) − u(i − 1)) − min ψ˜ ψ2 B− (v) = inf min N ∈N 2 i≤0 : u : −N ∪ {0} → R, u(i) = v(i) if i ≤ −N , Let v ± : Z → R. The transition energy between v − and v + is C(v − , v + ) = inf min N ∈N

ψ2

i∈Z

u(i + 2) − u(i)

+ψ 1 (u(i + 1) − u(i))− min ψ˜

2

: u : Z → R, c± ∈ R, u(i)=v ± (i) + c± if ± i ≥ N . +

−

−

+

Remark 5. Condition (H4) implies that C(uz , uz ) > 0 and C(uz , uz ) > 0 if z± ∈ M± . We can now describe the behaviour of minimizing sequences for (44). A more general statement can be found in [16]. Theorem 8. Suppose that (H1)–(H5) hold. We then have: (Case n even) The minimizers (un ) of (44) for n even converge, up to subsequences, to one of the functions x if 0 ≤ x ≤ L/2 −x if 0 ≤ x ≤ L/2 u+ (x) = u− (x) = L − x if L/2 ≤ x ≤ L, −(L − x) if L/2 ≤ x ≤ L. Let

+ + − − D := min B+ (uz ) + C(uz , uz ) + B− (uz ), −

−

B+ (uz ) + C(uz , uz ) + B− (uz ) : z+ ∈ M+ , z− ∈ M− }. +

+

If (un ) converges (up to subsequences) to u± then there exist z+ ∈ M+ , and z− ∈ M− such that +

+

−

−

D = B+ (uz ) + C(uz , uz ) + B− (uz ) and

(54)


En (un ) = D λn + o(λn ).

25

(55)

(Case n odd) In the case n odd the same conclusions hold, upon substituting terms of the form ±

±

∓

∓

±

±

∓

∓

B+ (uz ) + C(uz , uz ) + B− (uz ) by terms of the form B+ (uz ) + C(uz , uz ) + B− (uz ), where we have set (z1 , z2 ) = (z2 , z1 ). Proof. We only deal with the case n even, as the case n odd is dealt with similarly. Let un be a minimizer for (44). We may assume that un converge in W 1,p (0, L) and uniformly. By comparison with En (u) we have En (un ) ≤ L min ψ˜ + cλn .

(56)

We can consider the scaled energies En1 (u) =

1 ˜ (En (u) − L min ψ). λn

(57)

Note that we have En1 (u) =

n−2

ψ2

u

i=0

− ui 2λn

i+2

u 1 1 ui+2 − ui+1 i+1 − ui ψ + ψ1 − min ψ˜ 2 λn λn u − u 1 1 un − un−1 1 0 ˜ + ψ1 − min ψ. + ψ 2 λn λn +

(58)

From (56) and (58) we deduce that n−2

ψ2

i=0

+

1 1 ψ 2

u (xn ) − u (xn ) n i+2 n i 2λn u (xn ) − u (xn ) n

n

i+2

λn

i+1

+ ψ1

u (xn ) − u (xn ) n i+1 n i − min ψ˜ ≤ c. λn

We infer that for every η > 0 we have that if we denote by In (η) the set of indices i such that u (xn ) − u (xn ) n i+2 n i ψ2 2λn u (xn ) − u (xn ) 1 un (xni+2 ) − un (xni+1 ) n i+1 n i + ψ1 + ψ1 ≤ min ψ˜ + η 2 λn λn

26


then sup In (η) < +∞. n

Let ε = ε(η) be defined so that if z + z 1 1 2 ψ2 + ψ 1 (z1 ) + ψ 1 (z2 ) − min ψ˜ ≤ η 2 2 then dist ((z1 , z2 ), M) ≤ ε(η). Choose η > 0 so that 2ε(η) < min{|z+ − z− |, z+ ∈ M+ , z− ∈ M− }. We then deduce that if i − 1, i ∈ In (η) then there exists z ∈ M such that u (xn ) − u (xn ) u (xn ) − u (xn ) n i n i+2 n i+1 n i+1 − z ≤ ε , λn λn and

u (xn ) − u (xn ) u (xn ) − u (xn ) n i−1 n i+1 n i n i − z ≤ ε , λn λn Hence, there exist a finite number of indices 0 = i0 < i1 < i2 < · · · < iNn = n such that for all j = 1, . . . , Nn there exists znj ∈ M such that for all i ∈ {ij−1 + 1, . . . , ij − 1} we have u (xn ) − u (xn ) u (xn ) − u (xn ) n i n i+2 n i+1 n i+1 − znj ≤ ε. , λn λn Let {j0 , j1 , . . . , jMn } be the maximal subset of {i0 , i1 , . . . , iNn } defined by the requirement that if zjnk ∈ M± then zjnk +1 ∈ M∓ . Note that in this case we deduce that En (un ) ≥ cMn , so that Mn are equi-bounded. Upon choosing a subsequence we may then suppose Mn = M independent of n, and also that xnjk → xk ∈ [0, L] and znjk = zk . By the arbitrariness of η we deduce that limn un = u, and u is characterized by u(0) = u(L) = L and u = ±1 on (xk−1 , xk ), the sign determined by whether zk ∈ M+ or zk ∈ M+ . Let y0 = 0 < y1 < · · · < yN = L be distinct ordered points such that {yi } = {xk } (the set of indices may be different if xk = xk+1 for some k). Choose indices k1 , . . . , kN such that xnkj → (yj−1 + yj )/2. Let zj be the limit of znjk related to the interval (yj , yj+1 ). We then have, for a suitable continuous ω : [0, +∞) → [0, +∞), k 1 −2

u (xn ) − u (xn ) n i+2 n i 2λn u (xn ) − u (xn )

ψ2

i=0

1 1 n i+2 ψ 2 λn ≥ B+ (uz1 ) − ω(ε), +

n

i+1

+ ψ1

u (xn ) − u (xn ) n i+1 n i − min ψ˜ λn

From discrete systems to continuous energies kj+1 −2

ψ2

i=kj

+

27

u (xn ) − u (xn ) n i+2 n i 2λn

u (xn ) − u (xn ) 1 1 un (xni+2 ) − un (xni+1 ) n i+1 n i ψ + ψ1 − min ψ˜ 2 λn λn

≥ C(uzj , uzj+1 ) − ω(ε) for all j ∈ {1, . . . , N − 1}, n−2 i=kN

+

ψ2

u (xn ) − u (xn ) n i+2 n i 2λn

u (xn ) − u (xn ) 1 1 un (xni+2 ) − un (xni+1 ) n i+1 n i ψ + ψ1 − min ψ˜ 2 λn λn

≥ B− (uzN ) − ω(ε). By the arbitrariness of ε and the definition of D we easily get lim inf n En1 (un ) ≥ D, and by Remark 5 that if u = u± then lim inf n En1 (un ) > D. It remains to show that lim supn En1 (un ) ≤ D; i.e., for every fixed η > 0 to exhibit a sequence un such that un (0) = un (L) = 0 and lim supn En1 (un ) ≤ D + cη. Suppose that +

+

−

−

D = B+ (uz ) + C(uz , uz ) + B− (uz ), with z+ = (z1+ , z2+ ), z− = (z1− , z2− ), the other cases being dealt with in the same way. Let η > 0 be fixed and let N ∈ N, v+ , v− , v : Z → R be such that +

v+ (i) = uz (i)

for i ≥ N, v(i) =

+

uz (i) − uz (i)

−

v− (i) = uz (i)

for i ≤ −N,

for i ≤ −N for i ≥ N ,

and

ψ2

i≥0

ψ2

i≤0

ψ2

i∈Z

v (i + 2) − v (i) + + + + ψ 1 (u(i + 1) − u(i)) − min ψ˜ ≤ B+ (uz ) + η 2 v (i) − v (i − 2) − − − + ψ 1 (u(i) − u(i − 1)) − min ψ˜ ≤ B− (uz ) + η 2 v(i + 2) − v(i) 2

+ − + ψ 1 (v(i + 1) − v(i)) − min ψ˜ ≤ C(uz , uz ) + η.

28


We then set ⎧ (i) − v+ (0))λn ⎪ ⎪(v+ ⎪ ⎪ z+ n 1 n n ⎪ u ⎪ n (xi )− v+ (0)λn + zn (xi − xN ) ⎨ u(xni ) = v i − n2 λn − L2 ⎪ ⎪ − ⎪ ⎪ uzn (xnn−i ) − v− (0)λn + zn2 (xni − xnn−N ) ⎪ ⎪ ⎩ (v− (n − i) − v− (0))λn where

if i ≤ N if N ≤ i ≤ if

n 2 n 2

n 2

−N

−N ≤i≤

n 2

+N

if + N ≤ i ≤ n − N if n − N ≤ i ≤ n,

n L v+ (0)λn 2 λn − 2 + = n 2 − 2N λn − uz n2 λn + L2 + v− (0)λn n zn2 = . 2 − 2N λn +

zn1

uz

Note that limn zn1 = limn zn2 = 0. Using (H5) we easily get the desired inequality. Remark 6. From the proof above it can be easily seen that hypotheses (H3)– (H5) may be relaxed at the expense of a heavier notation and some changes in the results. Clearly, if (H3) does not hold then the sets of minimal pairs M+ , M− are larger, and the definition of D must be changed accordingly, possibly taking into account also more than one transition. + − − + If hypothesis (H4) does not hold then C(uz , uz ) = C(uz , uz ) = 0 for some z+ ∈ M+ , z− ∈ M− . In this case the energetic analysis of En1 is not sufficient to characterize the minimizers, as we have no control on the number of transitions between u = 1 and u = −1. Condition (H5) has been used to construct the recovery sequence (un ). It can be relaxed to assuming that ψ˜ is smooth at ±1; more precisely, it suffices to suppose that ˜ − min ψ˜ ψ(z) = 0. (59) lim z→±1 |z ∓ 1| If this condition does not hold the value D is given by a more complex formula, where we take into account also the values at 0 of the solutions of the boundary layer terms. The proof of Theorem 8 suggests the corresponding Γ -limit result for En1 . The details in a much more general setting can be found in [16]. 1.7 More examples In this section we examine some situations when some of the hypotheses considered hitherto are relaxed. Namely, (i) (weak nearest-neighbour interactions) when the condition ψn1 (z) ≥ c0 (|z|p − 1)


29

does not hold. In this case, the limit energy may be defined on a set of vector functions (multi-phase limit); (ii) (very-long-range interactions) when the energy En takes into account interactions up to the order Kn with Kn → +∞. In this case, the limit energy may be non-local; (iii) (non spatially homogeneous interactions) when the interaction between ui and ui+j may depend also on i. In this case a homogenization process may take place. For the sake of presentation we will explicitly treat only the case of quadratic energies, of the form En (u) =

Kn n−j

λn ρj,i n

j=1 i=0

u

− u i 2 , jλn

i+j

(60)

with ρj,i n > 0 and 1 ≤ Kn ≤ n. j 1 Remark 7. In the case when ρj,i n = ρ , Kn = K and ρ > 0 then the Γ -limit in Theorem 2 of En is given by

L

F (u) = ρ

|u |2 dt,

where ρ =

0

K

ρj .

j=1

j 1 The same conclusion holds if ρj,i n = ρ , Kn = n, ρ > 0, and ρ =

∞ j=1

ρj .

Weak nearest-neighbour interactions: multi-phase limits We only treat the case of next-to-nearest neighbour interactions with weak nearest-neighbour interactions; i.e., in (60) we take Kn = 2, ρ2,i n = c2 , and ρ1,i n = an with an lim 2 = c1 . n λn The energies we consider take the form En (u) = c2

n−2 i=0

λn

u

u 2 − u i 2 i+1 − ui + λn an . 2λn λn i=0 n−1

i+2

(61)

For all n un ∈ An (0, L), we consider un,e , un,o : {0, . . . , [n/2]} → R, defined by un,e (i) = un (2iλn ), un (xni )

un,o (i) = un ((2i + 1)λn )

= un (L) if i > n), which take into account the values (for simplicity, of un on even and odd points, respectively. Note that the energy En (un ) can be identified with an energy En (un,e , un,o ) defined by

30


[n/2]−1

En (un,e , un,o ) = c2

λn

u

n,e (i

i=0

[n/2]−1

+c2

λn

u

n,o (i

i=0

[n/2]−1

+

λn an

+ 1) − un,e (i) 2 2λn + 1) − un,o (i) 2 2λn

u

n,o (i)

− un,e (i) 2 λn

u

n,o (i)

− un,e (i + 1) 2 . λn

i=0

[n/2]−1

+

λn an

i=0

(62)

We say that the sequence (un ) converges (in L1 (0, L)) to u to the pair ñ,e , u ñ,o defined by (ue , uo ) if the piecewise-affine interpolates u u ñ,e =

un,e (i + 1) − un,e (i) 2λn

2i+2 on (x2i ), n , xn

un,o (i + 1) − un,o (i) 2i+2 on (x2i ), n , xn 2λn respectively, converge to (ue , uo ), respectively. We then have the following result. u ñ,o =

Theorem 9. The energies En Γ -converge with respect to the convergence of un to (ue , uo ), to the functional ⎧ L L L ⎪ 1 1 2 2 ⎪ ⎪ c c |u | dt + |u | dt + c |ue − uo |2 dt 2 1 ⎨2 2 e o 2 0 0 0 F (ue , uo ) = if ue , uo ∈ H 1 (0, L) ⎪ ⎪ ⎪ ⎩+∞ otherwise. If c1 = +∞ the formula above is understood to mean that F (ue , uo ) = +∞ if ue = uo , so that, having set u = ue = uo we recover for F the form ⎧ L ⎨ |u |2 dt if u ∈ H 1 (0, L) c2 F (u) = 0 ⎩ +∞ otherwise. Proof. It suffices to treat the case an = c1 λ2n with c1 < +∞, as all the others are easily obtained from that by a comparison argument. To obtain the liminf inequality, it suffices to use Theorem 1 for the first two terms in (62) and note that each of the last two terms converges to L 1 c1 |ue − uo |2 dt, 2 0 as the convergence of u ñ,e , u ñ,o to (ue , uo ), respectively, is uniform. The limsup inequality is obtained by direct computations on piecewiseaffine functions, and then reasoning by density as usual.


31

Very-long interactions: non-local limits For all n ∈ N let ρn : λn Z → [0, +∞). We consider the following form of the discrete energies

En (u) =

x,y∈λn Z∩[0,L] x=y

u(x) − u(y) 2 λn ρn (x − y) x−y

(63)

defined for u : λn Z → R. Note that we may assume that ρn is an even function, upon replacing ρn (z) by ρñ (z) = (1/2)(ρn (z) + ρn (−z)). We will tacitly make this simplifying assumption in the sequel. We will consider the following hypotheses on ρn : (H1) (equi-coerciveness of nearest-neighbour interactions) inf n ρn (λn ) > 0; (H2) (local uniform summability of ρn ) for all T > 0 we have ρn (x) < +∞. sup n

x∈λn Z∩(0,T )

Remark 8. Note that (H2) can be rephrased as a local uniform integrability property for λn ρn on R2 : for all T > 0 sup λn ρn (x − y) < +∞. n

x,y∈λn Z x=y,|x|,|y|≤T

As a consequence, if (H2) holds then, up to a subsequence, we can assume that the Radon measures λn ρn (x − y)δ(x,y) µn = x,y∈λn Z, x=y

(δz denotes the Dirac mass at z) locally converge weakly in R2 to a Radon measure µ0 , and that the Radon measures ρn (z)δz βn = z∈λn Z

locally converge weakly in R to a Radon measure β0 . These two limit measures are linked by the relation 1 |As |dβ0 (s), (64) µ0 (A) = √ 2 R where |As | is the Lebesgue measure of the set √ As = {t ∈ R : (s(e1 − e2 ) + t(e1 + e2 ))/ 2 ∈ A}. If (H1) holds then we have the orthogonal decomposition

32


β0 = β1 + c1 δ0 ,

(65)

for some c1 > 0 and a Radon measure β1 on R. We also denote µ = µ0

(R2 \ ∆)

(66)

(the restriction of µ0 to R2 \ ∆), where ∆ = {(x, x) : x ∈ R}. By the decomposition above, we have 1 µ0 = µ + √ c1 H1 2

∆,

where H1 stands for the 1-dimensional Hausdorff measure. The main result of this section is the following. Theorem 10 (Compactness and representation). If conditions (H1) and (H2) hold, then there exist a subsequence (not relabeled), a Radon measure µ on R2 and a constant c1 > 0 such that the energies En Γ -converge to the energy F defined on L1 (0, L) by ⎧ u(x) − u(y) 2 2 ⎨c1 |u | dt + dµ(x, y) if u ∈ W 1,2 (0, L) x−y F (u) = (0,L) (0,L)2 ⎩ +∞ otherwise, (67) wit respect to convergence in measure and L1 (0, L), where the measure µ and c1 are given by (66) and (65), respectively. Proof. Upon passing to a subsequence we may assume that the measures µn in Remark 8 converge to µ0 . Then, µ and c1 given by (66) and (65) are well defined. Hence, it suffices to prove the representation for the Γ -limit along this sequence. We begin by proving the liminf inequality. Let un → u in L1 (0, L) be such that supn En (un ) < +∞. By hypothesis (H1), the sequence un converges weakly in W 1,2 ((0, L). With fixed m ∈ N, we have the equality

En (un ) =

ρn (x − y)λn

x,y∈λn Z∩[0,L] |x−y|≤1/m, x=y

+

u (x) − u (y) 2 n n x−y

ρn (x − y)λn

x,y∈λn Z∩[0,L] |x−y|>1/m

=: In1 (un ) + In2 (un ). We now estimate these two terms separately.

u (x) − u (y) 2 n n x−y (68)


33

We first note that there exist positive αn converging to 0 such that

[αn /λn ]

lim 2 n

ρn (λn k) ≥ c1 −

k=1

1 . m

Let (a, b) ⊂ (0, L). For all N ∈ N and for n large enough we then have

In1 (un ) ≥

x,y∈λn Z∩(a,b) |x−y|≤αn , x=y

≥

≥

N

u (x) − u (y) 2 n n λn ρn (x − y) x−y

[αn /λn ]

i=1

k=1

x,y∈λn Z∩(yi−1 ,yi ) |x−y|=λn k

N

[αn /λn ]

i=1

2

2

u (x) − u (y) 2 n n λn ρn (λn k) x−y

u(y ) − u(y ) 2 (b − a) i i−1 ρn (λn k) +o(1) N yi − yi−1

k=1

as n → ∞, where we have set yi = a +

i (b − a), N

we have used the fact that un → u uniformly and the convexity of z → z 2 . This shows that 1 |u |2 dt. lim inf In1 (un ) ≥ c1 − n m (a,b) From this inequality we obtain that 1 |u |2 dt. lim inf In1 (un ) ≥ c1 − n m (0,L) As for the second term, for all η > 0 let ∆η = {(x, y) ∈ R2 : |x − y| > η}. Note that the convergence u(x) − u(y) un (x) − un (y) −→ x−y x−y is uniform on (0, L)2 \ ∆η , so that, by the weak convergence of µn we have u(x) − u(y) 2 lim inf In2 (un ) ≥ dµ(x, y). n x−y (0,L)2 \∆1/m By summing up all these inequalities and letting m → +∞ we eventually get

34


lim inf En (un ) ≥ c1 n

2

u(x) − u(y) 2

|u | dt +

x−y

(0,L)2

(0,L)

dµ(x, y).

To prove the limsup inequality it suffices to show it for piecewise-affine functions, since this set is strongly dense in the space of piecewise W 1,2 func tions. In this case it suffices to take un = u. Homogenization We only treat the case of nearest-neighbour interactions; i.e., in (60) we take Kn = 1 and ρ1,i n = ρi with i → ρi defining a M -periodic function Z → R: ρi+M = ρi . The energies we consider take the form En (u) =

n−1

λn ρi

u

i=0

i+1

− u i 2

λn

.

(69)

Theorem 11. The energies En Γ -converge to the energy defined by ⎧ L ⎨ ρ |u |2 dt if u ∈ H 1 (0, L) F (u) = 0 ⎩ +∞ otherwise, where ρ=M

M 1 −1 . ρ i=1 i

Proof. Note that M M ρ = min M ρi zi2 : zi = 1 , i=1

so that

i=1

M M ρz 2 = min M ρi zi2 : zi = z . i=1

i=1

We then immediately have n−1 i=0

λn ρi

u

i+1

− u i 2

λn

u

[n/M ]−1

≥

M λn ρ

i=0

− u M i 2 , M λn

M (i+1)

which gives the liminf inequality. The limsup inequality for the function u(x) = zx is obtained by choosing un defined by un (xni )

i 1 = ρzλn . ρk k=0


35

1.8 Energies depending on second-difference quotients We consider the case of energies En (u) =

n−1

λn f

u

i+1

i=1

− 2ui + ui−1 , λ2n

(70)

with f convex and such that c1 (|z|p − 1) ≤ f (z) ≤ c2 (1 + |z|p ) (p > 1). In this case we identify the discrete function u with a function in W 2,p (0, L). Given the values ui−1 , ui , ui+1 we define the function u on the interval Iin =

xn

i−1

+ xni xni+1 + xni n λn n λn , = xi − ,x + 2 2 2 i 2

(i ∈ {1, . . . , n − 1}) by u(t) =

xn + xni ui − ui−1 ui + ui−1 + t − i−1 2 λn 2 xni−1 + xni 2 ui+1 − 2ui + ui−1 t− + 2λ2n 2

Note that u =

ui+1 − 2ui + ui−1 λ2n

(71)

on Iin ,

and + xni ui + ui−1 = , 2 2 xn + xn u i+1 + ui i+1 = , u i 2 2

u

+ xni ui − ui−1 = 2 λn xn + xn u − ui i+1 i+1 u i = . 2 λn

xn

u

i−1

xn

i−1

Finally, we set u(t) =

u 1 − u0 λn u1 + u0 + t− 2 λn 2

on (0, λn /2) and u(t) =

un − un−1 λn un + un−1 + t−L− 2 λn 2

on (L − (λn /2), L). In this way u ∈ C 1 (0, L) and u is piecewise constant, so that u ∈ W 2,p (0, L) (actually, u ∈ W 2,∞ (0, L)). Moreover, En (u) = 0

We have the following result.

L

f (u ) dt.

(72)

36


Theorem 12. With the identification above, the energies En Γ -converge as n → +∞ to the functional ⎧ ⎨ f (u ) dt if u ∈ W 2,p (0, L) F (u) = (0,l) ⎩ +∞ otherwise with respect to the convergence in L1 (0, L) and weak in W 2,p (0, L). Proof. Let un → u in L1 (0, L) and supn En (un ) < +∞. Then we have sup n

L

(|un | + |un |p ) dt < +∞.

0

By interpolation, we deduce that supn un W 2,p (0,L) < +∞; hence un u weakly in u ∈ W 2,p (0, L). In particular un u in Lp (0, L), so that f (u ) dt ≤ lim inf f (un ) dt = lim inf En (un ). F (u) = n

(0,l)

(0,l)

n

If u ∈ C 2 ([0, L]) then, upon choosing (un )i = u(xni ) we have un → u and f (u + o(1)) dt, En (un ) = (0,l)

so that limn En (un ) = F (u). For a general u ∈ W 2,p (0, L) it suffices to use an approximation argument.

2 Limit energies on discontinuous functions: two examples In this chapter we begin dealing with energy density which do not satisfy a growth condition of polynomial type. We explicitly treat two model situations. 2.1 The Blake-Zisserman model A finite-difference scheme proposed by Blake-Zisserman to treat signal reconstruction problems takes into account (beside other terms of ‘lower order’) energies defined on discrete functions of the form En (u) =

n i=1

with

λ n ψn

u − u i i−1 , λn

α , ψn (z) = min z 2 , λn

(73)

(74)


37

for some α > 0. An interesting interpretation of the energy density ψn can be given also as relative to the energy between two neighbours in an array of material points connected by springs. In this case the springs are quadratic until a threshold, after which they bear no response to traction (broken springs). Note that the energies above do not fit in the framework of the previous chapter, as they do not satisfy a growth condition of order p from below. Note moreover that no interesting result can be obtained by taking into account the convexifications ψn∗∗ as they are trivially 0. In this section we will treat the limit of energies modeled on En above. We first define the proper convergence under which such energies are equicoercive. Coerciveness conditions We examine the coerciveness conditions for sequences of (piecewise-affine interpolations of) functions (un ) such that sup En (un ) < +∞.

(75)

n

For such a sequence, denote by I n = {i ∈ {1, . . . , n} : |un (xni ) − un (xni−1 )| > αλn }

(76)

the set of indices such that ψn (u ) = (u )2 and by

Sn =

on (xni−1 , xni ), (xni−1 , xni )

(77)

(78)

i∈I n

the union of the corresponding intervals. Note that we have (un )2 dt + α#(I n ), En (un ) =

(79)

(0,L)\Sn

so that by (75) we deduce that sup #(I n ) ≤ n

1 sup En (un ) < +∞. α n

(80)

Upon extracting a subsequence, we may assume then that #(I n ) = N

for all n ∈ N,

(81)

with N independent of n. Let tn0 , . . . , tnN +1 be points in [0, L] such that tn0 = 0, tnN +1 = L, tni−1 < tni and

38


{tni : i = 1, . . . , N + 1} = (λn I n ) ∪ {0, L}.

(82)

Upon further extracting a subsequence we may suppose that tni → ti ∈ [0, L]

for all i.

(83)

Denote the set of these limit points by S = {ti : i = 0, . . . , N + 1}. Let η > 0 be fixed; then for n large enough we have xni − (0, λn ) ⊂ S + (−η, η). Sn =

(84)

i∈I n

Hence, from (79) and (83) we deduce that (un )2 dt lim sup n (0,L)\(S+(−η,η)) ≤ sup (un )2 dt ≤ sup En (un ) < +∞. n

(0,L)\Sn

(85)

n

We deduce that for every η > 0 un ∈ W 1,2 ((0, L) \ (S + (−η, η))) and, if for every i = 0, . . . , N we have (86) lim inf ess-inf{|un (t)| : t ∈ (tni + η, tni+1 − η)} < +∞, n

then (un ) is weakly precompact in W 1,2 (tni + η, tni+1 − η) by Poincaré’s inequality. Let u be its limit defined separately on each (tni + η, tni+1 − η). By the arbitrariness of η we have that u can be defined on (0, L) \ S, and hence 1,p ((0, L) \ S). Moreover, by (85) we a.e. on (0, L). By this construction u ∈ Wloc deduce that for all η > 0 (u )2 dt ≤ lim inf (un )2 dt ≤ sup En (un ), (87) (0,L)\(S+(−η,η))

n

(0,L)\(S+(−η,η))

n

which gives a bound independent of η, so that by the arbitrariness of η > 0 we deduce that u ∈ W 1,p ((0, L) \ S). We now introduce the following notation. Definition 4. The space P-W 1,p (0, L) of piecewise-Sobolev functions on (0, L) is defined as the set of functions u ∈ L1 (0, L) such that a finite set S ⊂ (0, L) exists such that u ∈ W 1,p ((0, L) \ S). The minimal such set S is called the set of discontinuity points or jump set of u and denoted by S(u). For such u we regard the derivative u ∈ Lp (0, L) as defined a.e. and coinciding with its usual definition outside S. We then have the following compactness result.


39

Theorem 13. Let (un ) be a sequence of functions such that supn En (un ) < +∞ and such that (un ) is bounded in measure. Then there exists a function u ∈ P-W 1,2 (0, L) such that un → u in measure. Moreover there exists a finite 1,p ((0, L) \ S). set S such that un u weakly in Wloc Proof. The proof is contained in (76)–(87) above, once we remark that boundedness in measure implies (86). Limit energies for nearest-neighbour interactions From the reasonings above we easily deduce a first convergence result. Theorem 14. Let En be given by (73)–(74). Then En converge with respect to the convergence in measure and in L1 (0, L) to the energy ⎧ ⎨ F (u) =

L

|u |2 dt + α #(S(u))

⎩ 0 +∞

if u ∈ P-W 1,2 (0, L)

(88)

otherwise

in L1 (0, L). Proof. Let un → u in measure. Then by (79)–(87) it remains to show that #(S(u)) ≤ lim inf n #(I n ). This follows immediately from the facts that S(u) ⊂ S, and that, in the notation of (79)–(87), #(S) ≤ N = limn #(I n ). As for the limsup inequality, it suffices to remark that if we take un = u ∈ P-W 1,∞ (0, L) then for n large En (un ) ≤ F (u). For a general u we may proceed by density. From the lower semicontinuity properties of Γ -limits we immediately have the following corollary. Corollary 1. The functional F in (88) is lower semicontinuous with respect to the convergence in measure and in L1 (0, L). Remark 9. In Theorem 14 we can also consider the weak∗ -convergence of un . Equivalent energies on the continuum The first difference that meets the eye in Theorem 14 from the theory developed for energy densities with polynomial growth is that we have two different parts of the energy densities ψn that give rise to a bulk and a jump energy, respectively. In particular we cannot simply substitute the difference quotient by a derivative, or the function ψn by its convexification. A counterpart of En on the continuum is immediately obtained if we consider a different identification, other than the piecewise-affine one, for a discrete functions u : {xn0 , . . . , xnn } → R: using the notation

40


I n (u) = {i ∈ {1, . . . , n} : |u(xni ) − u(xni−1 )| > αλn }

(89)

we may extend u to the whole (0, L) by setting ⎧ ui − ui−1 ⎪ (t − xni ) if t ∈ (xni−1 , xni ), i ∈ I n (u) ⎪ ⎨ui−1 + λn u(t) = u if xni−1 ≤ x ≤ xni−1 + λ2n , i ∈ I n (u) i−1 ⎪ ⎪ ⎩ if xni − λ2n ≤ x ≤ xni , i ∈ I n (u). ui (90) Note that such extension of u belongs to P-W 1,2 (0, L), λn : i ∈ I n (u) , S(u) = xni − (91) 2 and we have the identification En (u) = F (u).

(92)

In this sense, F is the continuous counterpart of each En . Limit energies for long-range interactions We now investigate the limit of superpositions of energies of the form (73). Let (ρj ) and (αj ) be given sequences of non-negative numbers. We suppose that if αj ρj = 0 then αj = ρj = 0. We define the energy densities αj (93) ψnj (z) = min ρj z 2 , λn and the energies En (u) =

n n−j

λn ψnj

u

j=1 i=0

− ui jλn

i+j

(94)

Theorem 15. Suppose that ρ1 > 0,

α1 > 0.

(95)

Let ρ, α ∈ (0, +∞] be defined by ρ=

∞ j=1

ρj ,

α=

∞

jαj .

(96)

j=1

Then the energies En Γ -converge with respect to the convergence in measure and in L1 (0, L) to the functional F given by ⎧ L ⎨ |u |2 dt + α #(S(u)) if u ∈ P-W 1,2 (0, L) ρ F (u) = (97) 0 ⎩ +∞ otherwise in L1 (0, L), where it is understood that if α = +∞ then F (u) = +∞ if S(u) = ∅, and that if ρ = +∞ then F (u) = +∞ if u = 0 a.e.


41

Proof. Preliminarily note that by (95) we have that the Γ -limit (exists and) is +∞ outside P-W 1,2 (0, L). With fixed K ∈ N consider for n ≥ K the energies EnK (u) =

K n−j

λn ψnj

u

− ui , jλn

i+j

j=1 i=0

(98)

so that EnK (u) ≤ En (u).

(99)

For all j = 1, . . . , K and k = 0, . . . , j − 1 let

[n/j]−2

Enj,k (u) =

λn ψnj

u

k+(i+1)j

EnK (u) ≥

,

jλn

i=0

so that

− uk+ij

j−1 K

Enj,k (u).

(100)

(101)

j=1 k=0

Note that proceeding as in the proof of Theorem 14 by interpreting Enj,k as an energy on the lattice jλn Z + kλn , we easily get that Enj,k Γ -converge as n → +∞ to the functional F j (independent of k) given by ⎧ L ⎨ ρj |u |2 dt + αj #(S(u)) if u ∈ P-W 1,2 (0, L) j F (u) = (102) j 0 ⎩ +∞ otherwise. We then immediately get the following liminf inequality: if un → u then lim inf En (un ) ≥ lim inf EnK (un ) n

n

≥

j−1 K

lim inf Enj,k (un ) n

j=1 k=0

≥

K

jF j (u) =

j=1

K j=1

L

|u |2 dt +

0

K

jαj #(S(u)).

j=1

The desired inequality is obtained by letting K → +∞, and using the Monotone Convergence Theorem. Let now u ∈ P-W 1,2 (0, L) be such that F (u) < +∞. Consider first the case α < +∞, ρ < +∞. By a density argument it suffices to consider the case u ∈ P-W 1,∞ (0, L). In this case we can choose un = u, and note that lim sup En (un ) ≤ n

K j=1

j

j lim F (un ) + c n

∞

ρj u 2∞ + jαj #(S(u)) .

j=K+1

In the case when α = +∞ it suffices to compare with the convex case as ψnj (z) ≤ ρj z 2 . When ρ = +∞ F is finite only on piecewise-constant u, for which we take un = u and the computation is straightforward.

42


Boundary-value problems In contrast to what happened to functionals with limits defined on Sobolev spaces, the coerciveness conditions at our disposal do not guarantee that minimizers satisfying some boundary conditions converge to a minimizer satisfying the same boundary condition. We have thus to relax the notion of boundary values. We consider boundary-value problems given in two ways. (I) Interaction at the boundary: we fix two values U0 and UL and add to the energy En the constraint u(0) = U0 , u(L) = UL ; (II) Interaction through the boundary: we fix φ : R → R and add to En the ‘boundary-value term’ Bn (u) =

−1

n+K n

λn ψnj

j=1 i=max{−j,−Kn }

+

n+K n min{n+j,n+K n} j=1

u(xn ) − φ(xn ) i+j i jλn

λn ψnj

i=n+1

φ(xn ) − u(xn ) i i−j , jλn

which corresponds to setting u = φ outside [0, L] and to considering the energy of this extension on an enlarged interval with an addition of a ‘layer’ of size Kn λn on both sides of the interval. We first treat the case (II). For the sake of simplicity we consider the case when ρj = 0 for j > K, and we choose Kn = K. In this case our energy En + Bn can be written as ñ (u) = E

K n

λn ψnj

u

j=1 i=−j

− ui , jλn

i+j

(103)

with the constraint ui = φ(xin )

for i ∈ {−K, . . . , −1} ∪ {n + 1, . . . , n + K}

(104)

ñ be given by Theorem 16. Let φ : R → R be a continuous function. Let E ñ Γ - converges to the functional F˜ given by (103)–(104). Then E ⎧ L ⎪ ⎪ ⎪ρ |u |2 dt + α#({x ∈ [0, L] : u(x+) = u(x−)}) ⎨ 0 F˜ (u) = (105) if u ∈ P-W 1,2 (0, L) ⎪ ⎪ ⎪ ⎩+∞ otherwise, where ρ=

K

ρj ,

j=1

α=

K

αj ,

j=1

and we have set u(0−) = φ(0),

u(L+) = φ(L).

(106)


43

Proof. Let En (v, (−L, 2L)) be defined by En (v, (−L, 2L)) =

K 2n−j j=1 i=−n

λn ψnj

v

− vi . jλn

i+j

the choice of the interval (−L, 2L) has been done only for convenience of notation; indeed any open interval containing [0, L] would do. By the previous results En (·, (−L, 2L)) Γ -converges to the functional F (·, (−L, 2L)) with domain P-W 1,2 (−L, 2L) and defined there by 2L F (v, (−L, 2L)) = ρ |v |2 dt + α#(S(v)). −L

Let un → u in measure on (0, L). Let vn be defined by ⎧ ⎪ if i < 0 ⎨φ(0) n n vn (xi ) = un (xi ) if 0 ≤ i ≤ n ⎪ ⎩ φ(L) if i > n, and similarly define also v. Note that vn → v in measure on (−L, 2L). We then have ñ (un ) = lim inf En (vn , (−L, 2L)) lim inf E n

n

≥ F (v, (−L, 2L)) = F (u). To obtain the limsup inequality it suffices to take un = u. In the case (I) we treat arbitrarily long-range interactions. ñ be given by Theorem 17. Let En be given by (94) and let E ñ (u) = En (u) if u(0) = U0 and u(L) = UL E +∞ otherwise.

(107)

ñ Γ - converges to the functional F˜ given by Then E ⎧ L ⎪ ⎪ρ ⎪ |u |2 dt + α#(S(u)) + α0 #({x ∈ {0, L} : u(x+) = u(x−)}) ⎨ 0 ˜ F (u) = if u ∈ P-W 1,2 (0, L) ⎪ ⎪ ⎪ ⎩+∞ otherwise, (108) where ∞ ∞ ∞ ρj , α= jαj , α0 = αj , (109) ρ= j=1

j=1

j=1

and we have set u(0−) = U0 ,

u(L+) = UL .

(110)

44


Proof. We begin with the case j = 1 and with a boundary condition on only one side (e.g. at 0). Consider the functional En (u) if u(0) = U0 En0 (u) = +∞ otherwise. As in the proof of Theorem 16 we can write En0 (u) = En (v, (−L, L)),

where v(xni ) =

U0 u(xni )

and En (v, (−L, L)) =

n−1 i=−n

if i ≤ 0 if 0 < i ≤ n

λn ψn1

u

i+1

− ui

λn

.

If un → u we then obtain L lim inf En (un ) ≥ ρ1 |u |2 dt + α1 #(S(u)) + α1 (1 − χ0 (u(0+) − U0 )). n

0

The limsup inequality is immediately obtained by taking un = u. In the same way we treat the boundary condition at L and the boundary conditions at both sides. With fixed K can repeat the same reasoning as above for all j ∈ {1, . . . , K} such that αj ρj = 0 (otherwise the limit is trivial) and obtain that the Γ -limit of ⎧n−j ⎪ ⎨ λ ψ j ui+j − ui if u0 = U0 n n 0,j En (u) = i=0 λn ⎪ ⎩ +∞ otherwise as n → +∞ is given by ρj

L

|u |2 dt + jαj #(S(u)) + αj (1 − χ0 (u(0+) − U0 )).

0

Symmetrically we can treat the case un = UL . The case of boundary condition on both sides gives that the Γ -limit of ⎧n−j ⎪ ⎨ λ ψ j ui+j − ui if u(0) = U0 and un = UL n n 0,L,j En (u) = λn i=0 ⎪ ⎩ +∞ otherwise is


ρj

L

45

|u |2 dt + jαj #(S(u))

0

+αj (1 − χ0 (u(0+) − U0 )) + αj (1 − χ0 (u(L−) − UL )). Summing up these considerations we obtain that for all K

L

ñ ≥ ρK Γ - lim inf E n

|u |2 dt

0 K

+α #(S(u)) + α0K #({x ∈ {0, L} : u(x+) = u(x−)}), K K K where ρK = j=1 ρj , αK = j=1 jαj and α0K = j=1 αj . The liminf inequality is obtained by taking the supremum in K. The upper inequality is obtained by taking un = u. Remark 10. Note that we may have α = +∞ but α0 < +∞, in which case F˜ is finite only on W 1,2 (0, L) but may be finite also on functions not matching the boundary conditions. Homogenization We only treat the case of nearest-neighbour interactions. Let i → ρi and i → αi define M -periodic functions Z → R: ρi+M = ρi ,

αi+M = αi

for all i.

The energies we consider take the form En (u) =

u 2 i+1 − ui min λn ρi , αi . λn i=0

n−1

(111)

Theorem 18. The energies En Γ -converge to the energy defined by ⎧ L ⎨ ρ |u |2 dt + α#(S(u)) if u ∈ P-W 1,2 (0, L) F (u) = 0 ⎩ +∞ otherwise, where ρ=M

M 1 −1 , ρ i=1 i

α = min αi . i

Proof. By following the proof of Theorem 11 we immediately obtain the liminf inequality. In order to construct a recovery sequence for the Γ -limsup, let u ∈ t P-W 1,2 (0, L) and define v(t) = u(0+) + 0 u (s) ds. Let vn be a recovery sequence for the Γ -limit in Theorem 11 computed at v. Let k ∈ {0, . . . , M − 1}

46


be such that α = αk . Then for all t ∈ S(u) let jn (t) ≡ k mod M be such that |xnjn (t) − t| ≤ M λn . Then the functions

un (xni ) = vn (xni ) +

(u(t+) − u(t−))

t∈S(u): jn (t)≤i

define a recovery sequence for u. Non-local limits For all n ∈ N let ρn : jZ → [0, +∞). We consider the long-range discrete energies of Blake-Zisserman type

En (u) =

ρn (x − y) Ψn

u(x) − u(y)

x,y∈λn Z∩[0,L] x=y

x−y

(112)

defined for u : λn Z → R, where Ψn (z) = min{λn z 2 , 1}. We make the same assumptions on (ρn ) as in Section 1.7. Theorem 19. If conditions (H1) and (H2) in Section 1.7 hold, then there exist a subsequence (not relabelled), a Radon measure µ on R2 , a constant c1 > 0 and an even subadditive and lower semicontinuous function ϕ : R → [0, +∞] such that the energies En Γ -converge to the energy F defined on L1 (0, L) by ⎧ u(x) − u(y) 2 2 ⎪ ⎪ |u | dt + ϕ([u]) + dµ(x, y) c1 ⎪ ⎨ x−y (0,L) (0,L)2 S(u) F (u) = ⎪ if u is piecewise W 1,2 on [0, L] ⎪ ⎪ ⎩ +∞ otherwise, (113) where S(u) denotes the set of discontinuity points for u and [u](t) = u(t+) − u(t−) is the jump of u at t. The measure µ and c1 are given by (66) and (65), respectively, and the function ϕ is given by the discrete phase-transition energy density formula ϕ(z) = lim inf

inf lim min

m→+∞ |w| mλn mλn

:

(114)


47

Remark 11. (i) Since ϕ is subadditive, and it is also non decreasing on [0, +∞) and even, we have that either it is finite everywhere or ϕ(z) = +∞ for all z = 0. In the latter case jumps are prohibited and the domain of F is indeed W 1,2 (0, L). (ii) We will show below that the function ϕ may be not constant, in contrast with the case when ρn (z) = ρ(z/λn ) for a fixed ρ. Proof. With fixed m, n ∈ N the minimum value in (114) defines an even function of w which is non-decreasing on [0, +∞); hence, by Helly’s Theorem there exists a sequence (not relabeled) {λn } such that these minimum values converge for all w and for all m. Hence, we can assume, upon passing to this subsequence {λn }, that the function ϕ is well defined. Upon passing to a further subsequence we may also assume that the measures µn in Remark 8 converge to µ0 . Then, µ and c1 given by (66) and (65) are well defined as well. Hence, it suffices to prove the representation for the Γ -limit along this sequence, since the subadditivity and lower semicontinuity of ϕ are necessary conditions for the lower semicontinuity of F . We begin by proving the liminf inequality. Let un → u in L1 (0, L) be such that supn En (un ) < +∞. By hypothesis (H1), if we set S n = {x ∈ λn Z : |u(x + λn ) − u(x)|2 > 1/λn }, then #S n is equibounded, and, upon extracting a subsequence, we can suppose that S n = {xnj : j = 1, . . . , N } with N independent of n xn1 < xn2 < . . . < xnN and xnj → tj for all j. Set S = {tj } ⊂ [a, b]. If {xnM1 }, . . . , {xnM2 } are the sequences converging to t ∈ S then un (xnM1 ) → u(t−) and un (xnM2 + λn ) → u(t+). Furthermore, the sequence un converges locally weakly in W 1,2 ((0, L) \ S). For all η > 0 let Sη = {t ∈ R : dist (t, S) < η}; set also ∆η = {(x, y) ∈ R2 : |x − y| > η}. Note that the convergence u(x) − u(y) un (x) − un (y) −→ x−y x−y as n → ∞ is uniform on (0, L)2 \ (Sη2 ∪ ∆η ). With fixed m ∈ N, we have the inequality

En (un ) ≥

ρn (x − y)Ψn

x,y∈λn Z∩[0,L]∩S4/m |x−y|≤4/m, x=y

+

x,y∈λn Z∩[0,L]\S4/m |x−y|≤4/m, x=y

u (x) − u (y) n n x−y

ρn (x − y)Ψn

u (x) − u (y) n n x−y

48


+

ρn (x − y)Ψn

x,y∈λn Z∩[0,L] |x−y|>4/m

u (x) − u (y) n n x−y

=: In1 (un ) + In2 (un ) + In3 (un ).

(115)

The terms In2 (un ) and In3 (un ) can be dealt with as in Section 1.7. We now deal with In1 (un ). We first note that u (x) − u (y) n n . (116) In1 (un ) ≥ ρn (x − y)Ψn x−y t∈S(u) x,y∈λn Z∩[t−(2/m),t+(2/m)] x=y

We use the notation introduced above for the sets S n and S: let tj ∈ S(u) with corresponding sequences {xnM1 }, . . . , {xnM2 } converging to tj . We can suppose, up to a translation and reflection argument, that [u](tj ) > 0, that max{un (x) : x ∈ λn Z, tj − (2/m) ≤ x ≤ xnM1 } = 0 and that min{un (x) : x ∈ λn Z, xnM2 + λn ≤ x ≤ tj + (2/m)} = zn , with zn → [u](tj ). We then have u (x) − u (y) n n ρn (x − y)Ψn x−y x,y∈λn Z∩[tj −(2/m),tj +(2/m)] x=y

≥ min

x,y∈λn Z∩[tj −(2/m),tj +(2/m)] x=y

ρn (x − y)Ψn

v(x) − v(y) x−y

:

v(xnM1 ) = un (xnM1 ), v(xnM2 + λn ) = un (xnM2 + λn ) v(x) − v(y) : ≥ min ρn (x − y)Ψn x−y x,y∈λn Z∩[tj −(2/m),tj +(2/m)] x=y

v(x) = 0 if x ≤ xnM1 , v(x) = zn if x ≥ xnM2 + λn v(x) − v(y) ≥ min : ρn (x − y)Ψn x−y x,y∈λn Z∩[tj −(2/m),tj +(2/m)] x=y

2 1 1 2 v(x) = 0 if tj − ≤ x ≤ tj − , v(x) = zn if tj + ≤ x ≤ tj + m m m m v(j) − v(k) : ρn (λn (j − k))Ψn = min λn (j − k) j,k∈Z∩[−2/(mλn ),2/(mλn )] j=k

v(j) = 0 if −

1 1 2 2 .(117) ≤j≤− , v(j) = zn if ≤j≤ mλn mλn mλn mλn


49

Note that we have used the fact that Ψn in non decreasing on (0, +∞) so that our functionals decrease by truncation (namely, when we substitute v by (v ∨ 0) ∧ zn ). By taking (114) into account and summing up for tj ∈ S(u), we obtain ϕ([u](t)) + o(1) (118) lim inf In1 (un ) ≥ n

t∈S(u)

as m → +∞. By summing up this inequality to those obtained in Section 1.7 and letting m → +∞ we eventually get lim inf En (un ) ≥ c1 n

L

|u |2 dt +

0

u(x) − u(y) 2

ϕ([u]) + (0,L)2

S(u)

x−y

dµ(x, y).

We now prove the limsup inequality. It suffices to show it for piecewiseaffine functions, since this set is strongly dense in the space of piecewise W 1,2 functions. We explicitly treat the case when (0, L) is replaced by (−1, 1) and αt if t < 0 u(t) = βt + z if t > 0 only, as the general case easily follows by repeating the construction we propose locally in the neighbourhood of each point in S(u). It is not restrictive to suppose that z > 0, by a reflection argument, and that ϕ(z) < +∞, otherwise there is nothing to prove. Let η > 0, let m ∈ N with 0 < 1/m < η and let z − (1/m) < zm < z be such that u(j) − u(k) : ρn (λn (j − k))Ψn ϕ(z) ≥ lim min n λn (j − k) x,y∈Z,−2/(mλn )≤j,k≤2/(mλn ) 1 1 −η. (119) u : Z → R, u(j) = 0 if j < − , u(j) = zm if j > mλn mλn Then there exist functions vnm : λn Z → R such that vnm (x) = 0 for x < −1/m, vnm (x) = zm for x > T , 0 ≤ vnm ≤ zm and lim n

We set

ρn (x − y) Ψn

x,y∈λn Z −(2/m)≤x,y≤(2/m)

⎧ ⎪ ⎨u(t + (2/m)) (t) = um vnm (t) n ⎪ ⎩ u(t − (2/m))

v m (x) − v m (y) n

n

x−y

≤ ϕ(z) + η.

if t < −2/m if −2/m ≤ t ≤ 2/m if t > 2/m.

50


m Note that um in L1 ((−1, 1) \ [−1/m, 1/m]) as n → ∞, where n →u ⎧ u(t + (2/m)) if t < −2/m ⎪ ⎪ ⎪ ⎨0 if −2/m ≤ −1/m um (t) = ⎪ z if 1/m < t ≤ 2/m ⎪ ⎪ ⎩ u(t − (2/m)) if t > 2/m.

We can then easily estimate lim sup En (um n) n

≤ lim sup n

ρn (x − y) Ψn

x,y∈λn Z,x=y −2/m≤x,y≤2/m

+ lim sup (0,L)2 \∆2/m

n

+ lim sup n

n

n

x−y

um (x) − um (y) 1 n dµn Ψn n λn x−y um (x) − um (y) n ρn (x − y) Ψn n x−y

x,y∈λn Z∩[0,L], x,y1/m, |x−y|≤2/m

≤ ϕ(z) + η + (0,L)2

um (x) − um (y) 2 x−y

dµ + c1

um (x) − um (y) n

n

x−y

|u |2 dt + o(1)

(0,L)

as m → +∞. Note that we have used the fact that by (64) the limit measure µ does not charge ∂(0, L)2 . By choosing m = m(λn ) with m(λn ) → +∞ as m(λ ) n → ∞, and setting un = un n we obtain the desired inequality. In the following examples for simplicity we drop the hypothesis that ρn is even. Example 1. The function ϕ is not always constant. As an example, take ⎧ ⎪ 1 if z = λn ⎨√ √ ρn (z) = λn if z = λn [1/ λn ] ⎪ ⎩ 0 otherwise. Then it can be easily seen that the minimum for the problem defining ϕ is achieved on the function v = zχ(0,+∞) , which gives ϕ(z) = min{1 + z 2 , 2}. Note that in this case the Γ -limit is |u |2 dt + ϕ([u]), (0,L)

which is local, but not with ϕ constant.

S(u)


51

Example 2. If we take ⎧ ⎪ ⎨1√ ρn (z) = 4 λn ⎪ ⎩ 0

if z = λn √ if z = λn [1/ λn ] otherwise

then by using the (discretization of) v = zχ(0,+∞) as a test function we deduce the estimate ϕ(z) ≤ min{1 + 4z 2 , 5}. Since the right hand side is not subadditive, which is a necessary condition for lower semicontinuity, we deduce that the minimum in the definition of ϕ is obtained by using more than one ‘discontinuity’. Remark 12. By the density of the sums of Dirac deltas in the space of Radon measures on the real line, in the limit functional we may obtain any measure µ satisfying the invariance property µ(A) = µ(A + t(e1 + e2 )) for all Borel set A and t ∈ R. Remark 13. In the formula defining ϕ we cannot substitute the limit of minimum problems on [−2/(mλn ), 2/(mλn )] by a transition problem on the whole discrete line. In fact, if we take 1 if x = λn or x = λn [1/λn ] ρn (x) = 0 otherwise, then the two results are different. Example 3. By again √ taking ρn as in the previous remark, we check that in this case µ = (1/ 2)H1 (r1 ∪ r−1 ), where ri = {x − y = i}. 2.2 Second-difference interactions We may consider the same energies as in (73) but depending on seconddifference quotients; i.e., of the form En (u) =

n−1 i=1

λ n ψn

u

i+1

− 2ui + ui−1 , λ2n

(120)

with ψn as in (74). To understand the effect of the non-convexity threshold in ψn , first consider the case of a jump: ui = 0 for i ≤ i0 and ui = z = 0 for i > i0 . Then we have (for n large)

52


λ n ψn

u

− 2ui + ui−1 = λ2n

i+1

α 0

if i = i0 or i = i0 + 1 otherwise,

so that En (u) = 2α. If instead we have a crease; e.g., ui = 0 for i ≤ i0 and ui = cλn (i − i0 ) c = 0 for i > i0 , then we have u α if i = i0 i+1 − 2ui + ui−1 = λ n ψn λ2n 0 otherwise, so that En (u) = 2α. Mixing the arguments of Theorems 12 and 14 we obtain the following result (see [15]). Theorem 20 (jumps and crease limit energies). Let En be defined above. Then the domain of the Γ -limit F is the space of (possibly discontinuous) piecewise-W 2,2 functions, on which it takes the form L F (u) = |u |2 dt + 2α#(S(u)) + α#(S(u ) \ S(u)). 0

Note that the factor 2 in front of the ‘jump energy’ may be justified by relaxation arguments, as a jump point can be approximated by two ‘crease points’ (see [12] for the conditions for the lower-semicontinuity of this type of energies). 2.3 Lennard-Jones potentials We now consider a function J : R → R ∪ {+∞} modeling inter-atomic interactions, with the properties (i) J(z) = +∞ if z ≤ 0; (ii) J is smooth on (0, +∞); (iii) limz→0 J(z) = +∞. (iv) J is strictly convex on (0, T ); (v) J is strictly concave on (T, +∞); (vi) limz→+∞ J(z) = 0. Our assumptions are modeled on k1 k2 J(z) = 12 − 6 (121) z z for z > 0 All these conditions can be relaxed, and we refer to the general treatment in the next chapter for weaker assumptions. Note that hypotheses (ii)–(vi) imply that there exists a unique minimum point, which we denote by M ∈ (0, T ), and that min J < 0. The energy we will consider are, with fixed K ≥ 1, En (u) =

K n−j j=1 i=0

λn J

u

i+j

− ui

λn

.

(122)

Note the scaling in the argument of J; in terms of the general form considered in the previous chapter, we have ψnj (z) = J(jz).


53

Coerciveness conditions Note that En (u) is finite only if u is strictly increasing; hence, we can use the strong compactness properties of increasing functions. In particular, if (un ) is a sequence of functions locally equi-bounded on (0, L) then there exists a subsequence converging in L1loc (0, L), and if all functions are equi-bounded (e.g., if they satisfy some fixed boundary conditions) then there exists a subsequence converging in L1 (0, L) (actually, in Lp (0, L) for all p < ∞). Note moreover that, by Helly’s Theorem, upon passing to a further subsequence we can obtain convergence everywhere on (0, L). Nearest-neighbour interactions We begin by treating the case K = 1; i.e., En (u) =

n

λn J

i=1

u − u i i−1 . λn

(123)

It is easily seen that the Γ -limit is finite on all increasing functions. However, deferring the general treatment to the next chapter, we characterize the limit only on P-W 1,1 (0, L). Theorem 21. The energies En Γ -converge on P-W 1,1 (0, L) with respect to the L1 (0, L) convergence, to the functional F defined by ⎧ ⎨ ψ(u ) dt if u(t+) > u(t−) on S(u) F (u) = (124) (0,L) ⎩ +∞ otherwise on P-W 1,1 (0, L), where J(z) ψ(z) = J ∗∗ (z) = min J

if z ≤ M if z > M .

is the convex envelope of J. Note that the condition u(t+) > u(t−) on S(u) translates the fact that F must be finite only on increasing functions. Proof. Note preliminarily that F will be finite only on increasing functions so that we need to identify it only on functions u satisfying u(t+) > u(t−) on S(u). Recall that the functional ψ(u ) dt u → (a,b)

54


is lower semicontinuous on W 1,1 (a, b) with respect to the L1 (a, b) convergence. Let u ∈ P-W 1,1 (0, L) and write (0, L) \ S(u) =

N

(yk−1 , yk ),

(125)

k=1

where 0 = y0 < · · · < yN = L. Let un → u in L1 (0, L) and En (un ) < +∞ for all n. Then we have F (u) =

N k=1

ψ(u ) dt ≤

(yk−1 ,yk )

n

k=1

(yk−1 ,yk )

n

(0,L)

ψ(un ) dt

lim inf

ψ(un ) dt ≤ lim inf

≤ lim inf n

N

J(un ) dt = lim inf En (un ). n

(0,L)

Conversely, let u ∈ P-W 1,1 (0, L) with u(t+) > u(t−) on S(u), and let un = u. Then it is easily seen that J(u ) dt, lim En (un ) = n

(0,L)

so that

J(u ) dt.

Γ - lim sup En (u) ≤ n

(0,L)

Now, using the notation (125), let (ukj )j converge to u weakly in W 1,1 (yk−1 , yk ) (and hence also uniformly) and satisfy J((ukj ) ) dt = ψ(u ) dt. lim j

(yk−1 ,yk )

(yk−1 ,yk )

Note that for j sufficiently large the function uj defined by uj = ukj on (yk−1 , yk ) satisfies uj (t+) > uj (t−) on S(uj ) = S(u) and uj → u in L1 (0, L), so that, by the lower semicontinuity of the Γ -limsup we have J(uj ) dt Γ - lim sup En (u) ≤ lim inf Γ - lim sup En (uj ) ≤ lim inf j

n

=

N k=1

=

J((ukj ) ) dt

lim j

(yk−1 ,yk )

N

k=1

j

n

(yk−1 ,yk )

and the proof is concluded.

ψ(u ) dt = F (u),

(0,L)


55

Higher-order behaviour of nearest-neighbour interactions Note that minimum problems involving the limit functional F present a completely different behaviour depending on whether the (trivial) convexification of J is taken into account or not. Consider for example the simple minimum problem m = min F (u) : u(0) = 0, u(L) = h , (126) with h > 0. Then we have: (compression) if h ≤ M L then the minimum m = L J(h/L) is achieved only by the linear function u(x) = hx/L. Note that the minimizer has no jump; (tension) if h > M L then the minimum m = L min J is achieved by all functions u ∈ P-W 1,1 (0, L) such that u ≥ M a.e. Note in particular that we can exhibit minimizers with an arbitrary number of jumps. In this second case, hence, very little information on the behaviour of the minimizers of (127) mn = min En (u) : u(0) = 0, u(L) = h , can be drawn from the study of the corresponding problem (126) for the Γ -limit. To improve this description, we note now that minimizers of mn also minimize E (u) − L min J n : u(0) = 0, u(L) = h . (128) m(1) n = min λn The choice of the scaling λn is suggested by the fact that, choosing un = u, where u ∈ P-W 1,1 (0, L) is any function with u = M a.e. we have En (un ) ≤ L min J +cλn . We are then lead to studying the Γ -limit of the scaled functions En(1) (u) =

En (u) − L min J . λn

(129)

(1)

Theorem 22. The functionals En Γ -converge with respect to the L1 (0, L) convergence to the functional F (1) given by ⎧ 1,∞ ⎪ (0, L), u(t+) > u(t−) on S(u) ⎨− min J #(S(u)) if u ∈ P-W (1) F (u) = and u = M a.e. ⎪ ⎩ +∞ otherwise (130) on L1 (0, L) . Proof. Again, note preliminarily that F will be finite only on increasing functions so that we need to identify it only on functions u satisfying u(t+) > u(t−) on S(u).

56


The liminf inequality will be obtained by comparison. Let supn En (un ) < +∞ and un → u in L1 (0, L). Let vn (xni ) = un (xni ) − M xni . Note that vn → v = u − M x, and that ñ (vn ) = En (un ) = E

n

λ n ψn

i=1

where ψn (z) =

1 λn (J(z + M ) − min J).

v (xn ) − v (xn ) n i n i−1 , λn

Note that ψn → +∞ if z = 0, and that

lim ψn (z) = −

z→+∞

min J . λn

ñK be defined by With fixed k ∈ N let E ˜ K (w) = E n

v (xn ) − v (xn ) 2 1 1 n i n i−1 λn min k , (min J − ) . λn λn k i=1

n

ñK Γ -converge to F K defined by By the results of the previous chapter E ⎧ 1 ⎨k |w |2 dt + (min J − )#(S(w)) if w ∈ P-W 1,2 (0, L) K k F (w) = (0,L) ⎩ +∞ otherwise. ñK , so that Now, note that for n large enough we have En ≥ E ñ (vn ) ≥ lim inf E ñK (vn ) ≥ F K (v). lim inf En (un ) = lim inf E n

n

n

It will then be sufficient to consider the case v ∈ P-W 1,2 (0, L); that is, u ∈ P-W 1,2 (0, L). In this case we get 1 #(S(u)). lim inf En (un ) ≥ k |u − M |2 dt + min J − n k (0,L) By the arbitrariness of k we get the desired inequality. The limsup inequality is easily obtained. Indeed, if u ∈ P-W 1,∞ (0, L) is increasing and u = M a.e. we can take un = u, in which case En (un ) = L min J − min Jλn + o(λn ). Convergence of minimum problems From the results of the previous section we can easily derive a description of the limiting behaviour of minimizers of minimum problems (127). Proposition 2. Let h > M L; then from every sequence of minimizers of problems (127) we can extract a subsequence converging in L1 (0, L) to an increasing function u ∈ P-W 1,∞ (0, L) such that u = M a.e. in (0, L) and, after setting u(0−) = 0 and u(L+) = h, u has only one jump in [0, L]. Moreover we have the estimate mn = L min J − λn min J + o(λn ) as n → +∞.


57

Proof. By the coerciveness conditions on En , we can suppose that, upon extracting a subsequence, the minimizers of mn converge in L1 (0, L). We inter(1) pret those minimizers also as minimizers of mn . Hence, upon relaxing the boundary conditions, the limit function u solves the problem m(1) = − min J min{#(S(u)) : u ∈ P-W 1,∞ (0, L), u = M a.e., u(0−) = 0, u(L+) = h}. where S(u) is interpreted as a subset of [0, L]. The solution of this problem is clearly a function satisfying the thesis of the theorem, and m(1) = − min J. (1) From the convergence of minima (mn − L min J)/λn = mn → m(1) = − min J we complete the proof. Long-range interactions By taking into account the methods of Sections 1.4 and 1.4 and the proof of Theorem 21 we have the following result. Theorem 23. Let K ≥ 2 and let En be defined by En (u) =

K n−j j=1 i=0

λn J

u

i+j

− ui

λn

(131)

The energies En Γ -converge on P-W 1,1 (0, L) with respect to the L1 (0, L) convergence, to the functional F defined by ⎧ ⎨ ψ(u ) dt if u(t+) > u(t−) on S(u) (132) F (u) = (0,L) ⎩ +∞ otherwise on P-W 1,1 (0, L), where ψ : R → R ∪ {+∞} is the convex function given by ψ(z) = lim min N

−j K N 1 J(u(i + j) − u(i)) N j=1 i=0

u : {0, . . . , N } → R, u(i) = zi for i ≤ K or i ≥ N − K . (133) Furthermore, if K = 2 then the function ψ is also defined as ψ = J˜∗∗ , where ˜ = J(2z) + 1 min{J(z1 ) + J(z2 ) : z1 + z2 = 2z}. J(z) (134) 2 Proof. The proof follows by using the arguments of Theorems 21, 4 and 5, with ψnj (z) = J(jz), after noting that ψ defined above is convex and bounded at +∞. Remark 14. The function ψ satisfies the same assumptions as J upon replacing (vi) with limz→+∞ ψ(z) = C < 0, but it can be seen that J˜ in general does not satisfy (iv); i.e., is not of convex/concave form.

58


2.4 Renormalization-group approach A renormalization-group argument suggests a different scaling for LennardJones potentials. In the simplest case of nearest-neighbour interactions, we are led to consider the functionals En (u) =

n ui − ui−1 J M + λn − J(M ) . λn i=1

If J (M ) > 0 then the arguments for the first- and second-order Γ -limit above can be used simultaneously, to show the Γ -convergence to an energy whose domain are piecewise-H 1 functions with only increasing jumps, of the form F (u) = β

L

|u |2 dt + α#(S(u)),

u+ > u− on S(u),

0

where 2β = J (M ) and α = −J(M ). This result can be extended to longrange interactions with a formula for α that takes into account boundary-layer effects on the two sides of the jump (see [24]).

3 General convergence results In order to state and prove general results for the convergence of discrete schemes we will have to describe the Γ -limits of discrete energies in spaces of functions of bounded variation. We briefly recall some of their properties, referring to [8] for a complete introduction. 3.1 Functions of bounded variation We recall that the space BV (a, b) of functions of bounded variation on (a, b) is defined as the space of functions u ∈ L1 (a, b) whose distributional derivative Du is a signed Borel measure. For each such u there exists f ∈ L1 (a, b), a (at most countable) set S(u) ⊂ (a, b), a sequence of real numbers (at )t∈S(u) with t |at | < +∞ and a non-atomic measure Dc u singular with respect to the Lebesgue measure such that the equality of measures Du = f L1 + t∈S(u) at δt + Dc u holds. It can be easily seen that for such functions the left hand-side and right hand-side approximate limits u− (t), u+ (t) exist at every point, and that S(u) = {t ∈ R : u− (t) = u+ (t)} and at = u+ (t) − u− (t) =: [u](t). We will write u˙ = f , which is an approximate gradient of u. Dc u is called the Cantor part of Du. A sequence uj converges weakly to u in BV (a, b) if uj → u in L1 (a, b) and supj |Duj |(a, b) < +∞. The space SBV (a, b) of special functions of bounded variation is defined as = 0; i.e., whose distributhe space of functions u ∈ BV (a, b) such that Dc u tional derivative Du can be written as Du = u˙ L1 + t∈S(u) (u+ (t) − u− (t))δt .


59

This notation describes a particular case of a SBV -functions space as introduced by De Giorgi and Ambrosio [30]. We will mainly deal with functionals whose natural domain is that of piecewise-W 1,p functions, which is a particular sub-class of SBV (a, b) corresponding to the conditions u˙ ∈ Lp (a, b) and #(S(u)) < +∞, but we nevertheless use the more general SBV notation for future reference and for further generalization to higher dimensions (see [7]). For an introduction to BV and SBV functions we refer to the book by Ambrosio, Fusco and Pallara [8], while approximation methods for free-discontinuity problems are discussed by Braides [13]. A class of energies on SBV (a, b) are those of the form f (u) ˙ dt + g(u+ (t) − u− (t)), (a,b)

S(u)

with f, g : R → [0, +∞]. Lower semicontinuity conditions on F imply that f is lower semicontinuous and convex and g is lower semicontinuous and subadditive; i.e., g(x+y) ≤ g(x)+g(y). The latter can be interpreted as a condition penalizing fracture fragmentation, whereas convexity penalizes oscillations. If g is not lower semicontinuous and subadditive then we may consider its lower semicontinuous and subadditive envelope; i.e., the greatest lower semicontinuous and subadditive function not greater than g, that we denote by sub− g. For a discussion on the role of this condition for the lower semicontinuity of F we refer to [13] Section 2.2 or [14]. Energies in BV must satisfy further compatibility conditions between f and g (see e.g. Theorem 24 below and the subsequent remark). The following theorem is an easy corollary of [5] Theorem 6.3 and will be widely used in the next section. Theorem 24. For all n ∈ N let fn , gn : R → [0, +∞] be lower semicontinuous functions. Let α > 0 exists such that (1) fn is convex and for every z ∈ R,

α(|z| − 1) ≤ fn (z) (2) gn is subadditive and

for every z ∈ R.

α(|z| − 1) ≤ gn (z)

and suppose that f, g : R → [0, +∞] exist such that Γ - limn fn = f on R and Γ - limn gn = g on R \ {0}. For notation’s convenience we set g(0) = 0. Let Hn : BV (a, b) → [0, +∞] be defined as

Hn (u) :=

⎧ ⎪ ⎨ a

b

fn (u) ˙ dx +

⎪ ⎩ +∞

gn ([u])

if u ∈ SBV (a, b)

S(u)

otherwise.

60


Then Hn Γ -converge with respect to the weak topology of BV (a, b) to the functional H : BV (a, b) → [0, +∞) defined by l f (u) ˙ dx + σ + Dc u+ (a, b) + σ − Dc u− (a, b) + g([u]) H(u) := 0

(recall that Dc u Dc u), where

±

S(u)

denote the positive/negative part of the Cantor measure f (z) := inf{f (z1 ) + g 0 (z2 ) : z = z1 + z2 },

g(z) := inf{f ∞ (z1 ) + g(z2 ) : z = z1 + z2 }, z f (tz) f (±t) , g 0 (z) = lim tg , and σ ± = lim f ∞ (z) = lim t→+∞ t→+∞ t→+∞ t t t for all z ∈ R. Remark 15. Note that if we take gn = g and fn = f we recover the wellknown compatibility hypothesis f ∞ = g 0 for weakly lower semicontinuous functionals on BV (a, b). If f (0) = 0 then it can be easily seen that f = (f ∧ g 0 )∗∗ and g = sub− (f ∞ ∧ g). 3.2 Nearest-neighbour interactions For future reference, we state and prove the convergence results allowing for a more general dependence on the underlying lattice than in the previous chapters, at the expense of a slightly more complex notation. We begin by identifying the functions defined on a lattice with a subset of measurable functions. Consider an open interval (a, b) of R and two sequences (λn ), (an ) of positive real numbers with an ∈ [a, a + λn ) and λn → 0. For n n ∈ N let a ≤ x1n < . . . < xN n < b be the partition of (a, b) induced by the intersection of (a, b) with the set an + λn Z. We define An (a, b) the set of the restrictions to (a, b) of functions constant on each [a + kλn , a + (k + 1)λn ), k ∈ Z. A function u ∈ An (a, b) will be identified by Nn + 1 real numbers n c0n , . . . , cN n such that ⎧ i ⎪ if x ∈ [xin , xi+1 n ), i = 1, . . . , Nn − 1 ⎨cn 0 1 u(x) = cn (135) if x ∈ (a, xn ) ⎪ ⎩ Nn Nn if x ∈ [xn , b). cn For n ∈ N let ψn : R → [0, +∞] be a given Borel function and define En : L1 (a, b) → [0, +∞] as ⎧Nn −1 i ⎪ u(xi+1 ⎨ λ ψ n ) − u(xn ) x ∈ An (a, b) n n λn En (u) = (136) i=1 ⎪ ⎩ 1 +∞ otherwise in L (a, b). The following sections contain the description of the asymptotic behaviour of En as n → +∞.


61

Potentials with local superlinear growth We first treat the case when the potentials ψ satisfy locally a growth condition of order p > 1. This is the case of non-convex potentials introduced by Blake and Zisserman and of the scaled Lennard-Jones potentials which justify Griffith theory of fracture as a first-order effect. Theorem 25. For all n ∈ N let Tn± ∈ R exist with lim Tn± = ±∞,

lim λn Tn± = 0,

n

n

and such that, if we define fn , gn : R → [0, +∞] as ψn (z) if Tn− ≤ z ≤ Tn+ fn (z) = +∞ if z ∈ R \ [Tn− , Tn+ ] ⎧ z ⎨λ ψ z ∈ R \ [λn Tn− , λn Tn+ ] n n λ gn (z) = n ⎩ +∞ otherwise

(137)

(138)

(139)

the following conditions are satisfied: there exists p > 1 such that fn (z) ≥ |z|p gn (z) ≥ c > 0

∀z ∈ R ∀z = 0

(140) (141)

and, moreover, there exist f, g : R → [0, +∞], such that Γ - lim fn∗∗ = f on R,

(142)

Γ - lim sub− gn = g on R.

(143)

n

n

Then, (En )n Γ -converges to F with respect to the convergence in measure on L1 (a, b), where ⎧ ⎪ ⎨ F (u) =

a

b

f (u) ˙ dt +

⎪ ⎩ +∞

g([u](t))

u ∈ SBV (a, b)

t∈S(u)

otherwise in L1 (a, b).

Remark 16. Note that hypotheses (142) and (143) are not restrictive upon passing to a subsequence by a compactness argument. This remark also holds for Theorems 26 and 27. Moreover, if f is finite everywhere then Γ -convergence in (142) can be replaced by pointwise convergence. Proof. For simplicity of notation we deal with the case Tn+ = −Tn− =: Tn , the general case following by simple modifications. Without loss of generality we may assume

62


sup inf fn (z) < +∞; n z∈R

(144)

otherwise we trivially have f ≡ +∞ and consequently F ≡ +∞. With fixed u ∈ L1 (a, b) and a sequence (un ) ⊆ An (a, b) such that un → u in measure and supn En (un ) < +∞. Up to a subsequence, we can suppose in addition that un converges to u pointwise a.e. We now construct for each n ∈ N a function vn ∈ SBV (a, b) and a free-discontinuity energy such that vn still converges to u and we can use that energy to give a lower estimate for En (un ). Set i un (xi+1 n ) − un (xn ) (145) In := i ∈ {1, . . . , Nn − 1} : > Tn λn and ⎧ 1 ⎪ ⎪un (xn ) i+1 ⎨ (c − cin ) vn (x) := cin + n (x − xin ) ⎪ λ n ⎪ ⎩ un (x)

if x ∈ (a, x1n ) x ∈ [xin , xi+1 / In n ), i ∈

(146)

x elsewhere in (a, b).

We have that, for ε > 0 fixed, {x : |vn (x) − un (x)| > ε} i 1 ⊆ {x ∈ [xin , xi+1 / In , |un (xi+1 n ), i ∈ n ) − un (xn )| > ε} ∪ (a, xn ). (147) i Since, for i ∈ / In we have |un (xi+1 n ) − un (xn )| ≤ λn Tn , then {x : |vn (x) − un (x)| > ε} consists at most of the interval (a, x1n ) if n is large enough. Hence, the sequence (vn )n converges to u in measure and pointwise a.e. Moreover, by (141) (148) c#In ≤ En (un ) ≤ M,

with M = supn En (un ). By the equiboundedness of #In , we can suppose that S(vn ) = {xi+1 n }i∈In tends to a finite set. For the local nature of the arguments in the following reasoning, we can also assume that S consists of only one point x0 ∈ (a, b). Now, consider the sequence (wn )n defined by ⎧ ⎪ ⎪ v˙ n (t) dt if x < x0 ⎪ ⎨vn (a) + (a,x) wn (x) = (149) ⎪ v (a) + v ˙ (t) dt + [v ](t) if x ≥ x . ⎪ n n n 0 ⎪ ⎩ (a,x)

t∈S(vn )

Note that wn (a) = vn (a), w˙ n = v˙ n , S(wn ) = {x0 } and [wn ](x0 ) = t∈S(vn ) [vn ](t). Such a sequence still converges to u a.e. Indeed, since x0 is the limit point of the sets S(vn ), for any η > 0 fixed we can find n0 (η) ∈ N such that for any n ≥ n0 (η) and for any i ∈ In |x0 − xi+1 n | < η. Hence,


63

by construction, for any n ≥ n0 (η) and for any x ∈ (a, b) \ [x0 − η, x0 + η], wn (x) = vn (x), that is, the two sequences (vn ) and (wn ) have the same pointwise limit. Since w˙ n = v˙ n on (a, b), by (140) we have that w˙ n Lp (a,b) ≤ M . Then, using Poincaré’s inequality on each interval, it can be easily seen that (wn )n is equibounded in W 1,p ((a, b)\{x0 }). Since it also converges to u pointwise a.e., by using a compactness argument, we get that u ∈ W 1,p ((a, b)\{x0 }) and, up to subsequences, w˙ n u˙ weakly in Lp (a, b). Moreover, since for any two points a < x1 < x0 < x2 < b we have x2 wn (x2 ) = wn (x1 ) + w˙ n dt + [wn ](x0 ) x1

x2

u(x2 ) = u(x1 ) +

u˙ dt + [u](x0 ), x1

taking points x1 , x2 in which wn converges to u and passing to the limit as n → +∞, we have (150) [wn ](x0 ) → [u](x0 ). We can now rewrite our functionals in terms of vn : En (un ) = λn ψn (v˙ n ) + gn ([vn ](xi+1 n )) i∈I / n

i∈In

b

fn (v˙ n ) dt +

= a

gn ([vn ](t)).

t∈S(vn )

From (148) we also have

b

En (un ) ≥

fn (v˙ n ) dt + sub− gn

a

[vn ](t)

t∈S(vn ) b

≥

fn∗∗ (w˙ n ) dt + sub− gn ([wn ](x0 ))

a

as n → +∞. Passing to the liminf as n → +∞, using (150) we have lim inf En (un ) ≥ lim inf n

≥

n

b

fn∗∗ (w˙ n ) dt + lim inf sub− gn ([wn ](x0 ))

a

n

b

f (u) ˙ dt + g([u](x0 )) a

as desired. We now turn our attention to the construction of recovery sequences for the Γ -limsup. We may assume in what follows that inf z∈R fn (z) = fn (0).

64


Step 1 We first prove the limsup inequality for u affine on (a, b). Set ξ = u; ˙ we can assume, upon a slight translation argument, that f (ξ) = limn fn∗∗ (ξ). Then, for each n in N we can find ξn1 , ξn2 ∈ R, tn ∈ [0, 1] such that √ λn 1 2 |tn ξn + (1 − tn )ξn − ξ| ≤ 2(b − a) tn fn (ξn1 ) + (1 − tn )fn (ξn2 ) ≤ fn∗∗ (ξ) + o(1) |ξni | ≤ c = c(ξ).

(151)

Note that in the last inequality the choice of the constant c can be chosen independent of n thanks to (140) and (144). It can be easily seen that it is not restrictive to make the following assumptions on ξni : ξn1 > ξ, fn (ξn1 ) ≤ fn (ξn2 ), (|ξn1 | + |ξn2 |) λn ≤ 1. (152) We define a piecewise-affine function vn ∈ L1 (a, b) with the following properties: vn (x) = u(x) on (a, x1n ],

v˙ n

| [xin , xi+1 n )

:= vni ∈ {ξn1 , ξn2 },

and vni is defined recursively by ⎧ vn1 = ξn1⎧ ⎪ ⎪ ⎪ ⎨ i √ ⎪ λn ⎨v i if ≤ vn (an ) + vnj λn + vni λn − u(xi+1 λn n n )≤ 2 i+1 ⎪ vn = j=1 ⎪ ⎪ ⎪ ⎩ ⎩ξ 1 + ξ 2 − v i otherwise. n n n (153) √ Since 0 ≤ vn − u ≤ λn by definition, (vn )n converges to u uniformly, and hence in measure and, moreover, " ! (154) βn1 := # i ∈ {0, . . . , Nn } : vni = ξn1 ≥ tn Nn . Indeed, from (151), (152) and (153) we deduce √ λn Nn n ≤ vn (xN λn Nn tn (ξn1 − ξ) + (1 − tn )(ξn2 − ξ) ≤ n ) − u(xn ) 2 = βn1 (ξn1 − ξ)λn + (Nn − βn1 )(ξn2 − ξ)λn , so that (βn1 − tn Nn )(ξn1 − ξn2 ) ≥ 0. Now, consider the sequence (un ) ⊆ An (a, b) defined by un (xin ) = vn (xin ) for i = 1, . . . , Nn ,

un (a) = vn (a) and un (b) = vn (b).

Since (147) still holds with un , vn as above, it can be easily checked that (un )n converges to u in measure. Hence, recalling (151), (152) and (154),


65

En (un ) = λn fn (ξn1 )βn1 + (Nn − βn1 )λn fn (ξn2 ) ≤ tn λn Nn fn (ξn1 ) + (1 − tn )Nn λn fn (ξn2 ) ≤ Nn λn (fn∗∗ (ξ) + o(1)) ≤ (b − a)fn∗∗ (ξ) + o(1). Taking the limsup as n → +∞ we get lim sup En (un ) ≤ f (ξ)(b − a) = F (u). n

The same construction#as above works also in the case of a piecewise-affine function: let [a, b] = [aj , bj ] with a1 = a, bj = aj+1 and u˙ constant on each (aj , bj ), then it suffices to repeat the procedure above on each (aj , bj ) to provide functions vnj in An (aj , bj ) such that vnj → u lim sup n

| (aj , bj )

in measure λ n ψn

{i: xin ∈(aj ,bj )}

j i vnj (xi+1 n ) − vn (xn ) λn

≤

bj

f (u) ˙ dx. aj

With j fixed define ynj := max{xin ∈ (aj , bj )}. Then, the recovery sequence un is defined in (aj , bj ) as un (x) = vnj (x) − (vn+1 (yn + λn ) − vn (yn )). <j

λn

√ Since u(x) + 2 ≤ vnj (x) ≤ u(x) + λn by construction, and |u(yn + λn ) − u(yn )| ≤ cλn , we have that un → u in measure and j i+1 vn (xn ) − vnj (xin ) + cfn (0)λn . λ n ψn En (un ) = λn i j {i: xn ∈(aj ,bj )}

By a density argument we can extend the result to functions in W 1,p (a, b). Step 2 Let u be of the form zχ(x0 ,b) with g(z) < +∞ and let zn be a recovery sequence for g(z) = Γ -limn sub− gn (u). The sequence sub− gn (zn ) is bounded, hence, by (141), upon possibly considering a suitable subsequence, there exists an integer N not depending on n such that N $ N − i i gn (z ) : z − zn < ε . sub gn (zn ) = sup inf ε i=1

i=1

Hence, for all n we can find N points {zn1 , . . . , z N } such that lim n

N i=1

zni = z and lim n

N i=1

gn (zni ) = g(z).

(155)

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Let in ∈ {1, . . . , Nn } be the index such that x0 ∈ [xinn , xnin +1 ) and, for n large, define wn as in (135) with ⎧ 0 if i ≤ in ⎪ ⎪ ⎪ j ⎪ ⎨ zn if in in + N . ⎩ j=1

Clearly (wn )n → u in measure and En (wn ) =

N

λ n ψn

i=1

zni λn

=

N

gn (zni ) + (b − a)fn (0) ;

i=1

the estimate follows from (155) by passing to the limit as n → +∞. Step 3 Let u ∈ SBV (a, b) be such that F (u) < +∞, then

x

u˙ dt + c and w(x) =

u = v + w with v(x) = a

m

zj χ[xj ,b) .

j=1

# i+i Nj For j = 1, . . . m let wnj be defined as in Step 2 with jumps in j {xn n,j }i=1 and let vn be a recovery sequence for v such that it is constant on each m i i wnj converges in measure to [xnn,j , xnn,j + λn Nj ). The sequence un = vn + j=1

u and m En (wnj ) ≤ F (v) + F (w) = F (u), lim sup En (un ) = lim sup En (vn ) + n

n

j+1

as desired. Corollary 2. Let ψn : R → [0, +∞] satisfy the hypotheses of Theorem 25. Assume that, in addition, for all n ∈ N, fn = fn∗∗ on [Tn− , Tn+ ] and gn = sub− gn on R \ [λn Tn− , λn Tn+ ]. Then, for any u ∈ L1 (a, b), En (u) Γ -converges to F (u) with respect to the strong topology of L1 (a, b). Proof. It suffices to produce a recovery sequence converging strongly in L1 (a, b). Note that in Step 1, by the convexity of fn , we can choose ξn1 = ξn2 = ξn in (151). Then vn = u and un turns out to be the piecewise-constant interpolation of u at points {xin }. It is easy to check that un → u strongly in L1 (a, b). It remains to show that also for functions of the form zχ[x0 ,b) it is possible to exhibit a sequence that converges strongly in L1 (a, b). To this end it suffices to note that in Step 2, since gn = sub− gn locally on R \ {0}, we can find a sequence (zn ) such that (155) is replaced by limn zn = z and limn gn (zn ) = g(z). Hence, the sequence wn defined by (156) converges to u strongly in L1 (a, b) and it is a recovery sequence.


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Remark 17. Note that the hypotheses of the previous corollary are satisfied if ψn is convex and lower semicontinuous on [Tn− , Tn+ ] and concave and lower semicontinuous on (−∞, Tn− ] and [Tn+ , +∞) Potentials with linear growth In this section we will consider energy potentials ψn such that ψn (z) ≥ α(|z| − 1)

∀z ∈ R

(157)

for some α > 0. For this kind of energies we can still prove a convergence result to a free-discontinuity energy, whose volume and surface densities are obtained by a suitable interaction of the limit functions f , g of the two ‘regularized’ scalings of ψn . Note that in the following statement the sequences Tn± are arbitrary. Theorem 26. Let ψn : R → [0, +∞] satisfy (157). For all n ∈ N let Tn± ∈ R satisfy properties (137) and let fn , gn : R → [0, +∞] be defined as in (138). Assume that f, g : R → [0, +∞] exist such that Γ - lim fn∗∗ = f on R,

(158)

Γ - lim sub− gn = g on R \ {0}.

(159)

n

n

For notation’s convenience we set g(0) = 0. Then, (En )n Γ -converges to F with respect to the convergence in L1 (a, b) and the convergence in measure, where ⎧ b ⎪ − ⎪ ⎪ f (u) ˙ dx + g([u]) + c1 Du+ ⎪ c (a, b) + c−1 Duc (a, b) ⎨ F (u) =

a

⎪ ⎪ ⎪ ⎪ ⎩+∞

S(u)

if u ∈ BV (a, b) otherwise, f (z) := inf{f (z1 ) + g 0 (z2 ) : z1 + z2 = z}, g(z) := inf{f ∞ (z1 ) + g(z2 ) : z1 + z2 = z},

∞

∞

c1 := f (1) and c−1 := f (−1). Remark 18. Thanks to (157) the theorem can be restated also with respect to the weak convergence in BV (a, b). Indeed, sequences converging in measure along which the functionals En are equibounded are weakly compact in BV . Proof. Again we deal with the case Tn+ = −Tn− =: Tn , the general case being achieved by slight modifications. Let un , u ∈ L1 (a, b) be such that un → u in measure and En (un ) ≤ c. Analogously to the proof of Theorem 25, we will estimate En (un ) by a free-discontinuity energy computed on a sequence vn

68


converging to u weakly in BV (a, b). Let In and vn be defined as in (145) and (146), respectively. Note that vn → u in measure and that vn has equibounded total variation on (a, b). Indeed, by hypothesis (157) we have |Dvn |(a, b) =

N n −1

i |un (xi+1 n ) − un (xn )| ≤

i=1

1 En (un ) + c. α

From this inequality we easily get that u ∈ BV (a, b), in particular the Γ -liminf is finite only on BV (a, b). Up to passing to a subsequence we may assume that vn converges to u weakly in BV (a, b); moreover, by construction we have b fn∗∗ (v˙ n ) dt + sub− gn ([vn ]). En (un ) ≥ a

S(vn )

Hence, it suffices to apply Theorem 24 to the functionals on the left hand side to get the Γ -liminf inequality. To obtain the converse inequality it suffices to provide a sequence vn converging to u in L1 (a, b) such that b lim sup En (vn ) ≤ F1 (u) := f (u) ˙ dt + g([u]) n

a

S(u)

when u ∈ SBV (a, b). The general estimate will be then obtained by relaxation (i.e. by taking fn = f and gn = g in Theorem 24). By a standard approximation argument it is sufficient to prove this inequality in the simpler cases of u linear and of u with a single jump. Let u(t) = ξt; we may assume that f (ξ) < +∞. Moreover, we may assume in what follows that inf z∈R fn (z) = fn (0). Then we can find ξn1 , ξn2 such that the analogue of (151) holds. In this case, |ξn1 |, |ξn2 | are not necessarily equibounded; nevertheless we have by definition |ξn1 |, |ξn2 | ≤ Tn since f (ξ) < +∞. Thus we can construct the of Theorem 25, up to a functions vn as in the proof of the Γ -limsup-inequality √ slight modification. Indeed, if we replace λn with λn Tn in (151) and (153), all those inequalities still hold. In particular we have that |u(x) − vn (x)| ≤ λn Tn in (a, b). Thus vn is a recovery sequence converging to u in L∞ (a, b). Mn i zn be such As for the case of u = zχ(x0 ,b) with g(z) < +∞, let zn = i=1 M n Mn i that g(z) = limn i=1 gn (zni ). Note that since limn i=1 zn = z, by taking Mn i + (157) into account, we may assume that i=1 (zn ) ≤ c|z| and the same for the negative terms. We may assume also that |zni | ≥ λn Tn for any i. Hence, by arguing separately on the positive and negative part of (zni ), we easily get that c|z| . (160) Mn ≤ λn Tn Finally we can construct a sequence of functions wn defined as in (156) where we replace N with Mn . By taking (160) into account we easily get


|{x : wn (x) = u(x)}| ≤ λn Mn ≤

69

c|z| . Tn

Then wn → u in L∞ (a, b) and, by construction, lim sup En (wn ) = g(z) = F1 (u). n

The desired upper estimate follows then by standard arguments.

Potentials of Lennard-Jones type We now treat the case of potentials with non-symmetric growth conditions, which still ensure weak-BV compactness of sequences with equibounded energies. These conditions are satisfied for example by Lennard-Jones potentials. Theorem 27. Let ψn : R → [0, +∞] satisfy ψn (z) ≥ (|z|p − 1)

for all z < 0.

(161)

for some p > 1. For all n ∈ N let Tn ∈ R satisfy lim Tn = +∞, n

lim λn Tn = 0,

(162)

n

and let fn , gn : R → [0, +∞] be defined by ψn (z) z ≤ Tn fn (z) = +∞ z > Tn+ ⎧ z ⎨λ ψ if z > λn Tn n n λn gn (z) = ⎩ +∞ otherwise.

(163)

(164)

Assume that there exist f, g : R → [0, +∞] such that Γ - lim fn∗∗ = f on R,

(165)


(166)

n

n

For notation’s convenience we set g(0) = 0. Then, (En )n Γ -converges to F with respect to the convergence in L1loc (a, b) and the convergence in measure, where ⎧ b ⎪ ⎪ − ⎪ f (u) ˙ dx + g([u]) + σDu+ ⎪ c (a, b) if u ∈ BVloc (a, b) Dc u = 0 ⎨ F (u) =

a

⎪ ⎪ ⎪ ⎪ ⎩+∞

S(u)

and [u] > 0 on S(u) otherwise in L1 (a, b). ∞

where f and g are defined as in Theorem 26 and σ := f (1).

70


Proof. Let un → u in measure and be such that En (un ) ≤ c, and assume that un → u also pointwise. Set i In = {i ∈ {1, . . . , Nn } : un (xi+1 n ) − un (xn ) > λn Tn },

and let vn be the sequence of functions defined as in (146) with this choice of In . Note that vn → u in measure. By taking hypothesis (161) into account we have the following estimate on the negative part of the (classical) derivative of vn b (un (xi+1 ) − un (xi ))− p En (un ) n n + c. |(v˙ n )− |p dt ≤ λn ≤ λ α n a i∈In

Hence, with fixed δ > 0 and with fixed x1 , x2 points in (a, a + δ), (b − δ, b), respectively, in which vn converges pointwise to u, we get b b−δ (v˙ n )+ dt + [vn ] ≤ |vn (x2 ) − vn (x1 )| + (v˙ n )− dt. a+δ

a

S(vn )∩(a+δ,b−δ)

It follows that vn is bounded in BVloc (a, b). Since vn → u in measure we get that vn converges in BVloc (a, b) to u and hence u ∈ BVloc (a, b). With fixed η > 0, consider fnη (z) = (fn )∗∗ (z) + η|z|,

gnη (z) = sub− gn (z) + η|z|.

For every δ ∈ (0, (b − a)/2) we have b−δ En (un ) + ηc(δ) ≥ fnη (v˙ n ) dt + a+δ

gnη ([vn ]),

S(vn )∩(a+δ,b−δ)

where c(δ) = supn |Dvn |(a + δ, b − δ). We can apply Theorem 24 and obtain for every η and δ lim inf En (un ) + ηc(δ) n b−δ ∞ ≥ f (u) ˙ dt + f (1)|Dc u+ |(a + δ, b − δ) + a+δ

g([u]).

S(u)∩(a+δ,b−δ)

By letting η → 0 and subsequently δ → 0 we obtain the desired inequality. The construction of a recovery sequence for the Γ -limsup follows the same procedure as in the proof of Theorem 26. Examples Example 4. (i) (Blake-Zisserman model) The typical example of a sequence of functions which satisfy the hypotheses of Theorem 25 (and indeed of Corollary 2) is given (fixed (λn ) converging to 0 and C > 0) by


ψn (z) =

1 (λn z 2 ) ∧ C), λn

with p = 2, Tn = C/λn , |z| ≤ C/λn z2 fn (z) = +∞ otherwise, so that

71

gn (z) =

C +∞

|z| > C/λn |z| ≤ C/λn ,

b

|u| ˙ 2 dt + C#(S(u))

F (u) = a

on SBV (a, b) (ii) Theorem 25 allows also to treat asymmetric cases. As an example, let 1 (λn z 2 ) ∧ C) if z > 0 ψn (z) = λ2n if z ≤ 0. z In this case the Γ -limit (with respect to both the convergence in measure and L1 convergence) is given by ⎧ ⎨ F (u) =

⎩

Note that

b

|u| ˙ 2 dt + C#(S(u))

if u ∈ SBV (a, b) and u+ > u− on S(u)

a

+∞

otherwise. ⎧ ⎪ ⎨C g(z) = 0 ⎪ ⎩ +∞

if z > 0 if z = 0 if z < 0

forbids negative jumps. (iii) (Lennard-Jones type potentials) Let ψ : R → [0, +∞] be a lower semicontinuous function and satisfy ψ(z) = 0 if and only if z = 0, ψ(z) ≥ α(|z|p − 1) for z < 0 and limz→+∞ ψ(z) = C. Let ψn = ψ for all n. Then we can apply Theorem 27 and obtain f = ψ ∗∗ and 0 if z ≥ 0 g(z) = +∞ if z < 0. Note that f (z) = 0 if z ≥ 0. (iv) (scaled Lennard-Jones type potentials) Let ψ be as in the previous example, and choose 1 ψn (z) = ψ(z). λn Then we can apply Theorem 25 with

72


0 f (z) = +∞

if z = 0 otherwise,

and g as in Example (ii) above. In this case the limit energy F is finite only on piecewise-constant functions with a finite number of positive jumps. On such functions F (u) = C#(S(u)). We now give an example which illustrates the effect of the operation of the subadditive envelope. Example 5. If we take ψn (z) = z 2 ∧

1 √ + (|z| λn − 1)2 λn

with λn converging to 0, then we obtain f (z) = z 2 and z2 : k = 1, 2, . . . . g(z) = sub− (1 + z 2 ) = min k + k A remark on second-neighbour interactions Consider functionals of the form u(xi+1 ) − u(xi ) u(xi+2 ) − u(xi ) n n n n En (u) = + . (167) λn ψn1 2λn ψn2 λ 2λ n n i i If both sequences of functions (ψni )n satisfy conditions of Corollary 2 and some additional growth conditions from above, then it can be seen that the conclusions of Theorem 25 hold with f (z) = lim ψn1 (z) + 2ψn2 (z) , n

and

z z + 4ψn2 . g(z) = lim λn ψn1 n λn 2λn

This means that En can be decomposed as the sum of three ‘nearest-neighbour type’ functionals, with underlying lattices λn Z, 2λn Z and λn (2Z + 1), respectively, whose Γ -convergence can be studied separately. We now show that a similar conclusion does not hold if we remove the convexity/concavity hypothesis on ψni . Example 6. Let (λn ) be a sequence of positive numbers converging to 0, and let M > 2 be fixed. Let En be given by (167) with 2 √ z if |z| ≤ 1/ kλn k √ ψn (z) = 1 k kλn z if |z| > 1/ kλn kλn g


73

(k = 1, 2), where 1

g (z) =

M 1

if |z| < 8 if |z| ≥ 8

Neither g i is subadditive and we have 2 if |z| < 8 sub− g 1 (z) = 1 if |z| ≥ 8

2

g (z) =

1 M

if |z| ≤ 1 if |z| > 1.

1 sub− g 2 (z) = 2

if |z| ≤ 1 if |z| > 1.

We can view En as the sum of a first-neighbour interaction functional and two second-neighbour interaction functionals, to whom we can apply separately Theorem 25, obtaining the limit functionals b |u| ˙ 2 dt + sub− g 1 ([u]) F 1 (u) = a

S(u)

for the first, and

b

|u| ˙ 2 dt +

F 2 (u) = a

sub− g 2 ([u])

S(u)

for each of the second ones. We will show that the Γ -limit of En is strictly greater than F 1 (u) + 2F 2 (u) at some u ∈ SBV (a, b). Let u be given simply by u = χ(t0 ,b) with t0 ∈ (a, b). In this case F 1 (u) + 2 2F (u) = 4. Suppose that there exist un ∈ An (a, b) converging to u and such that lim supn En (un ) ≤ 4. In this case it can be easily seen that for n large enough there must exist in such that un (xin ) − un (xin −1 ) > 4,

un (xin +1 ) − un (xin ) < −4,

but |un (xin −1 ) − un (xin −2 )| < 1,

|un (xin +2 ) − un (xin +1 )| < 1.

This implies that un (xin ) − un (xin −2 ) > 3,

un (xin +2 ) − un (xin ) < −3,

so that lim supn En (un ) ≥ 2M , which gives a contradiction. 3.3 Long-range interactions We conclude this chapter with a general statement whose proof can be obtained by carefully using the arguments of Section 1.4 and of the previous sections in this chapter. We use the notation of Chapter 1.

74


Let K ∈ N be fixed. For all n ∈ N and j ∈ {1, . . . , K} let ψnj : R → (−∞, +∞] be given Borel functions bounded below. Define En : L1 (0, L) → [0, +∞] as ⎧ K n−j i ⎪ ⎪ u(xi+j n ) − u(xn ) j ⎨ x ∈ An (0, L) λ n ψn jλn (168) En (u) = j=1 i=0 ⎪ ⎪ ⎩+∞ 1 otherwise in L (0, L). We will describe the asymptotic behaviour of En as n → +∞ when the energy densities are potentials of Lennard-Jones type. More precisely, we will make the following assumptions. (H1) (growth conditions) There exists a convex function Ψ : R → [0, +∞] and p > 1 such that Ψ (z) = +∞ lim z→−∞ |z| and there exist constants c1j , c2j > 0 such that c1j (Ψ (z) − 1) ≤ ψnj (z) ≤ c2j max{Ψ (z), |z|} for all z ∈ R. Remark 19. Hypothesis (H1) is designed to cover the case of Lennard-Jones potentials (and potential of the same shape. Another case included in hypotheses (H1) is when all functions satisfy a uniform growth condition of order p > 1; i.e., (|z|p − 1) ≤ ψnj (z) ≤ c(|z|p + 1) for all j and n. Before stating our main result, we have to introduce the counterpart of the energy densities fn and gn in the previous section for the case K > 1. The idea is roughly speaking to consider clusters of N subsequent points (N large) and define an average discrete energy for each of those clusters, so that the energy En may be approximately regarded as a ’nearest neighbour interaction energy’ acting between such clusters, to which the above description applies. We fix a sequence (Nn ) of natural numbers with the property lim Nn = +∞, n

lim n

Nn = 0. n

(169)

We define ψn (z) = min

K N n −j u(i + j) − u(i) 1 : u : {0, . . . , Nn } → R, ψnj Nn j=1 i=0 j (170) u(x) = zx if x = 0, . . . , K, Nn − K, . . . , Nn .


75

By using the energies ψn we will regard a system of Nn neighbouring points as a single interaction between the two extremal ones, up to a little error which is negligible as Nn → +∞. We can now state our convergence result, whose thesis is exactly the same as that of Theorem 27 upon replacing λn by εn := Nn λn . Theorem 28. Let ψnj satisfy (H1) and let (En ) be given by (168). Let ψn be given by (170) and let εn = Nn λn . For all n ∈ N let Tn ∈ R be defined as in (162), and let fn , gn : R → [0, +∞] be defined by ψn (z) z ≤ Tn fn (z) = (171) +∞ z > Tn ⎧ z ⎨ε ψ if z > εn Tn n n εn gn (z) = (172) ⎩ +∞ otherwise. Assume that there exist f, g : R → [0, +∞] such that Γ - lim fn∗∗ = f on R,

(173)


(174)

n

n

Note that this assumption is always satisfied, upon extracting a subsequence. Then, (En )n Γ -converges to F with respect to the convergence in L1loc (0, L) and the convergence in measure, where ⎧ L ⎪ ⎪ f (u) ˙ dx + g([u]) + σDu+ if u ∈ BVloc (0, L) Dc u− = 0 ⎪ c (0, L) ⎨ 0 S(u) F (u) = ⎪ and [u] > 0 on S(u) ⎪ ⎪ ⎩ +∞

otherwise in L1 (0, L).

where f and g are defined by (for notational convenience we set g(0) = 0) f (z) := inf{f (z1 ) + g 0 (z2 ) : z1 + z2 = z}, g(z) := inf{f ∞ (z1 ) + g(z2 ) : z1 + z2 = z}, ∞

and σ := f (1).

References 1. R. Alicandro, A. Braides and M. Cicalese. Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint. Networks and Heterogeneous Media 1 (2006), 85–107.

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2. R. Alicandro, A. Braides and M. Cicalese. Continuum limits of discrete thin films with superlinear growth densities. Preprint SNS, Pisa, 2005. 3. R. Alicandro and M. Cicalese. Representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math Anal. 36 (2004), 1–37. 4. R. Alicandro, M. Focardi and M.S. Gelli, Finite-difference approximation of energies in fracture mechanics. Ann. Scuola Norm. Pisa (IV) 29 (2000), 671709. 5. M. Amar and A. Braides, Γ -convergence of nonconvex functionals defined on measures, Nonlinear Anal. TMA 34 (1998), 953-978. 6. L. Ambrosio, Existence theory for a new class of variational problems, Arch. Rational Mech. Anal. 111 (1990), 291-322. 7. L. Ambrosio and A. Braides, Energies in SBV and variational models in fracture mechanics. In Homogenization and Applications to Material Sciences, (D. Cioranescu, A. Damlamian, P. Donato eds.), GAKUTO, Gakk¯ otosho, Tokio, Japan, 1997, p. 1-22. 8. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, Oxford, 2000. 9. A. Blake and A. Zisserman, Visual Reconstruction, MIT Press, Cambridge, Massachusetts, 1987. 10. X. Blanc, C. Le Bris and P.-L. Lions. From molecular models to continuum models. Arch. Rational Mech. Anal. 164 (2002), 341–381. 11. X. Blanc, C. Le Bris and P.-L. Lions. Atomistic to continuum limits for Computational Materials Science. Math. Mod. Numer. Anal., to appear. 12. A. Braides, Lower semicontinuity conditions for functionals on jumps and creases. SIAM J. Math Anal. 26 (1995), 1184–1198. 13. A. Braides, Approximation of Free-Discontinuity Problems, Lecture Notes in Mathematics 1694, Springer Verlag, Berlin, 1998. 14. A. Braides, Γ -convergence for Beginners, Oxford University Press, Oxford, 2002. 15. A. Braides. Discrete approximation of functionals with jumps and creases, in Homogenization 2001., GAKUTO, Gakkotosho, Tokyo, 2003, 147–154. 16. A. Braides and M. Cicalese, Surface energies in nonconvex discrete systems. Preprint SNS, Pisa, 2004. 17. A. Braides, G. Dal Maso and A. Garroni, Variational formulation of softening phenomena in fracture mechanics: the one-dimensional case, Arch. Rational Mech. Anal. 146 (1999), 23-58. 18. A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Oxford University Press, Oxford, 1998. 19. A. Braides and G.A. Francfort. Bounds on the effective behavior of a square conducting lattice. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460 (2004), 1755–1769. 20. A. Braides and M.S. Gelli, Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids, vol. 7, 1, (2002), 41-66. 21. A. Braides and M.S. Gelli, Limits of discrete systems with long-range interactions, Journal of Convex Analysis, vol. 9, 2 (2002), 363-399. 22. A. Braides and M.S. Gelli, The passage from discrete to continuous variational problems: a nonlinear homogenization process, in “Nonlinear homogenization and its applications to composites, polycrystals and smart materials (P. Ponte Castaneda, J.J. Telega and B. Gambin eds.), Kluwer, 2004, 45–63.


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23. A. Braides, M.S. Gelli and M. Sigalotti, The passage from non-convex discrete systems to variational problems in Sobolev spaces: the one-dimensional case, Proceedings of the Steklov Institute of Mathematics, 236 (2002), 408-427. 24. A. Braides, A.J. Lew and M. Ortiz. Effective cohesive behavior of layers of interatomic planes. Arch. Ration. Mech. Anal., 180 (2006), 151–182. 25. A. Braides and A. Piatnitski. Overall properties of a discrete membrane with randomly distributed defects. Preprint SNS, Pisa, 2004. 26. A. Chambolle. Un theoreme de Γ -convergence pour la segmentation des signaux. C. R. Acad. Sci., Paris, Ser. I 314 (1992), 191-196. 27. A. Chambolle. Image segmentation by variational methods: Mumford and Shah functional and the discrete approximations. SIAM J. Appl. Math. 55 (1995), 827–863. 28. A. Chambolle. Finite-differences discretizations of the Mumford-Shah functional. M2AN Math. Model. Numer. Anal. 33 (1999), 261–288. 29. G. Dal Maso, An Introduction to Γ -convergence, Birkh¨ auser, Boston, 1993. 30. E. De Giorgi and L. Ambrosio, Un nuovo funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 82 (1988), 199-210. 31. G. Del Piero and L. Truskinovsky. A one-dimensional model for localized and distributed failure, Journal de Physique IV France 8, 8 (1998), 95-102. 32. G. Friesecke and F. Theil. Validity and failure of the Cauchy-Born hypothesis in a two-dimensional mass-spring lattice. J. Nonlinear Sci. 12 (2002), 445–478. 33. B. Houchmandzadeh, J. Lajzerowicz and E. Salje, Relaxation near surfaces and interfaces for first-, second- and third-neighbour interactions: theory and applications to polytypism, J. Phys.: Condens. Matter 4 (1992), 9779-9794. 34. C. Le Bris and P.-L. Lions. From atoms to crystals: a mathematical journey. Bull. Amer. Math. Soc. 42 (2005), 291–363. 35. D. Mumford and J. Shah, Optimal approximation by piecewise smooth functions and associated variational problems, Comm. Pure Appl. Math. 17 (1989), 577685. 36. S. Pagano and R. Paroni. A simple model for phase transitions: from the discrete to the continuum problem. Quart. Appl. Math. 61 (2003), 89–109. 37. A. Piatnitski and E. Remy. Homogenization of elliptic difference operators. SIAM J. Math. Anal. 33 (2001), 53–83. 38. J. Pouget, Dynamics of patterns in ferroelastic-martensitic transformations, Phys. Rev. B 43 (1991), 3575-3581. 39. L. Truskinovsky, Fracture as a phase transition, Contemporary research in the mechanics and mathematics of materials (R.C. Batra and M.F. Beatty eds.) CIMNE, Barcelona, 1996, 322-332.

Relaxation for bulk and interfacial energies Anneliese Defranceschi Dipartimento di Matematica, Universit` a di Trento Via Sommarive, 14, Povo, Trento, Italy e-mail: [email protected]

1 Introduction Several physical phenomena in phase transitions, fracture mechanics, image segmentation may be formulated by a variational model; this implies the study of the existence of equilibria, and hence of the existence of minimizers, for the related energy functional. This will be done by using the so called direct method of the Calculus of Variations based on lower semicontinuity and coerciveness of the energy functional. It is clear that more general lower semicontinuous results allow to consider more general variational models, and when the lower semicontinuity is not available, the characterization of the associated relaxed functional allows to understand the behaviour of minimizing sequences of the functional we started with. In these lectures we will present some lower semicontinuity and relaxation results for functionals of the form f (∇u) dx + g(u+ , u− , ν) dH n−1 , (1) F (u) = Ω

Su ∩Ω

where u is a “smooth” function outside a (n − 1)-dimensional discontinuity set Su on which the traces u+ and u− and the normal ν are well defined. Our main goal will be (i) the search for (necessary and) sufficient conditions on f and g which will guarantee lower semicontinuity of the functional F in some natural function space; (ii) the study of the relaxed energy, and the identification of the relaxed functional in some integral form, when the lower semicontinuity of F fails. Minimum problems associated to (1) are named, after De Giorgi, “free discontinuity problems” since together with a standard volume part we have a surface energy term, concentrated on (n−1)-dimensional sets. An important feature of these problems lies in the fact that the support of the surface energy is not fixed a priori, and is usually an unknown of the problem.

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Anneliese Defranceschi

Let us mention just two of the principal examples of variational problems that can be described in this setting. Let us consider an elastic body which under the action of force fields or subjected to boundary displacements may fracture in some (a priori unknown) region. Here Ω ⊆ Rn is a bounded open domain (n = 3 in the applications) and represents the reference configuration of the body, and u : Ω → Rn stands for the displacement field. The corresponding energy functional is of the form (1): f (∇u) is the bulk energy density relative to the elastic deformation outside the fracture, whereas g(u+ , u− , ν) is the surface energy density on the fracture Su (necessary to produce the crack). In the framework of Griffith’s materials, the energy necessary to the production of the crack is proportional to the crack surface, that is, g(u+ , u− , ν) = α ; in this case (1) becomes of the special form f (∇u) dx + αH n−1 (Su ∩ Ω) . F (u) = Ω

Different models have been proposed to allow an interaction between the two sides of the crack opening; this can be done by considering, in the isotropic case, functionals like f (∇u) dx + g(|u+ − u− |) dH n−1 F (u) = Ω

Su ∩Ω

with g(t) → 0 as t → 0 (Barenblatt’s theory). A second important example of functional of the form (1) is the leading part of the weak formulation in SBV of the Mumford-Shah functional for the segmentation problem given by |∇u|2 dx + αH n−1 (Su ∩ Ω) + β (u − g)2 dx . G(u) = Ω\Su

Ω

As above Ω is a bounded open set in Rn , α and β are fixed positive parameters, and g ∈ L∞ (Ω). When n = 2 and g represents the 2-dimensional image given by a camera, then the minimization problem associated to G corresponds to the search for a “smoothed” approximated image u outside the set of contours Su . For a detailed discussion of the previous functional and other examples of variational problems that can be formulated and studied in this setting we refer to the book by [13], Section 4 and Section 6. Let us briefly describe the functional framework we have to consider in order to define in a correct way the functional (1). The natural function space of weakly differentiable functions allowing the presence of discontinuity sets is the space BV (Ω; Rk ) of functions of bounded variation. For every u ∈ BV (Ω; Rk ) the quantities Su , u+ , u− , . . . involved in our above description

Relaxation for bulk and interfacial energies

81

(see Sections 4 and 7) are well defined. Moreover, the measure Du can be decomposed as Du = ∇uLn + (u+ − u− ) ⊗ νH n−1

Su + Dc u ,

where ∇u denotes the density of the absolutely continuous part of Du with respect to the Lebesgue measure, (u+ − u− ) ⊗ νH n−1 Su is the Hausdorff part (jump part) and Dc u is the so called Cantor part of Du, concentrated on a Ln negligible set and such that (Dc u)(B) = 0 whenever H n−1 (B) < +∞. We denote by SBV (Ω; Rk ) the class of special functions of bounded variation, i.e., u belongs to SBV (Ω; Rk ) if u ∈ BV (Ω; Rk ) and Dc u = 0. This latter space is the right environment to define the functional (1). However, there are significant cases (e.g., if g(a, b, ν) behaves like a multiple of |a − b|, for a close to b) where the domain of the relaxed functional is larger than SBV (see, for the 1-dimensional case Theorem 13, for then n-dimensional case Theorem 17, Theorem 21 and Theorem 25). Let us come now to the plan of these lectures. In the first part (corresponding to Sections 2-4) we recall the basic notation and give the main ingredients we will use afterwards, such as lower semicontinuity, the direct method of the Calculus of Variations and relaxation, as well as the notion and properties of functions of (special) bounded variation. Then we state some suitable versions of general lower semicontinuity theorems and prove relaxation results for functionals of type (1) in dimension 1. These will be subsequently extended to higher dimension as well as for vector-valued functions. Finally, let us point out that in the 1-dimensional case the relaxation results can be proved ’by hand’ whereas in the higher dimensional case one has to use fine properties of BV functions as the co-area formula. Since such formula is not available for vector-valued BV -functions, we have to proceed in a completely different and more technical manner in order to proof the main relaxation theorem of Section 7 (see also Section 8).

2 Lower semicontinuity - The direct method of the Calculus of Variations The direct method of the Calculus of Variations can be summarized in the equation Lower Semicontinuity + Coerciveness =⇒ Existence of minimizers meaning that, in order to obtain the existence of minimizers for a certain functional, it is sufficient to find a suitable topology in which a compact set

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can be found where a minimizing sequence lies (coerciveness), and at the same time the functional under consideration is lower semicontinuous. In this section we recall briefly the notions of lower semicontinuity and coerciveness, and state the direct method of the Calculus of Variations (for more details and proofs we refer to the books [33],[27],[23]). Let X be a metric space, and let R = R ∪ {−∞, +∞}. Let F : X → R be a function; we use the notation {F ≤ t} = {x ∈ X : F (x) ≤ t}. The level sets {F < t}, {F ≥ t}, {F > t} are defined in a similar way. Lower semicontinuity Definition 1. A function F : X → R is said to be (sequentially) lower semicontinuous (l.s.c. for short) at x, if F (x) ≤ lim inf F (xh ) h→+∞

for every sequence (xh ) converging to x, or in other words F (x) = min{lim inf F (xh ) : xh → x} . h→+∞

We will say that F is lower semicontinuous (on X) if it is lower semicontinuous at all x ∈ X. Remark 1. (i) If F and G are l.s.c. at x, then so is F + G. (ii) Let {Fi : i ∈ I} be an arbitrary family of l.s.c. functions. Then F (x) = sup Fi (x) is l.s.c. i∈I

(iii) If F = 1E is the characteristic function of the set E (see (6)) then F is l.s.c. if and only if E is open. Coerciveness condition Definition 2. We say that a subset K of X is (sequentially) compact if every sequence in K has a subsequence which converges to a point of K, i.e., ∀(xh ) ⊂ K

∃x ∈ K ,

∃(xhk ) : xhk → x .

Definition 3. A function F : X → R is (sequentially) coercive if {F ≤ t} is compact in X for every t ∈ R, i.e., the closure of {F ≤ t} is compact in X for every t ∈ R. We are now in a position to describe Tonelli’s direct method for proving existence results in the Calculus of Variations. Let F : X → R be a function. A minimum point (or minimizer) for F in X is a point x ∈ X such that F (x) ≤ F (y) for every y ∈ X, i.e., F (x) = inf F (y) . y∈X


83

A minimizing sequence for F in X is a sequence (xh ) in X such that inf F (y) = lim F (xh ) .

y∈X

h→+∞

Direct method - (Weierstrass theorem) Theorem 1. Let F : X → R be coercive and lower semicontinuous. Then (i) F has a minimum point in X. (ii) If (xh ) is a minimizing sequence of F in X, and x is the limit of a subsequence of (xh ), then x is a minimum point of F in X. (iii) If F is not identically +∞, then every minimizing sequence for F has a convergent subsequence.

3 Relaxation Let F : X → R be a coercive functional, but not lower semicontinuous. In general we can not say that F has a minimum point on X. Therefore we study the notion of relaxation, which allows to describe the minimizing sequences of functionals that are not lower semicontinuous in terms of minimum points of suitable lower semicontinuous functionals; in other words, we associate to F a functional denoted by F , which admits a minimum on X and has the following properties: (i) min F (x) = inf F (x) ; x∈X

x∈X

(ii) the minimum points for F are the limits of minimizing sequences of F , and every minimizing sequence of F has a subsequence which converges to a minimum point of F . Definition 4. Let F : X → R. The lower semicontinuous envelope (or relaxed function) F of F , denoted also by sc− F , is the greatest lower semicontinuous function less than or equal to F , i.e., for every x ∈ X F (x) = sup{G(x) : G : X → R , G is l.s.c. , G ≤ F } .

(2)

By Remark 1 (ii) we have that (2) gives a lower semicontinuous function. Let us point out that the function F can be characterized in terms of sequences as follows: for every x ∈ X ! " F (x) = min lim inf F (xh ) : xh → x " ! h→+∞ (3) = min lim F (xh ) : xh → x , and ∃ lim F (xh ) . h→+∞

h→+∞

We consider now the connection between the minimum problem min F (x) x∈X

and the relaxed problem min F (x). In particular, the following theorem dex∈X

scribes the behaviour of the minimizing sequences of F in terms of the minimizers of F .

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Theorem 2. Let F : X → R be coercive. Then the following properties hold: (i) F is coercive and lower semicontinuous; (ii) F has a minimum point in X ; (iii) min F (x) = inf F (x) ; x∈X

x∈X

(iv) the minimum points for F are exactly all the limits of minimizing sequences of F . Remark 2. For functionals of the form f (x, u, ∇u) dx F (u) =

(4)

Ω

defined on some Sobolev space W 1,p (Ω) (p ≥ 1) the direct method introduced in Section 2 is useless whenever the integrand function f has linear growth of the type |z| ≤ f (x, u, z) ≤ c(1 + |z|) . (5) Indeed, even if f satisfies all assumptions of the lower semicontinuity theorems, the coerciveness condition fails because the sets " ! |∇u| dx ≤ t u ∈ W 1,p (Ω) : Ω

are not sequentially relatively compact on W 1,p (Ω) endowed with its weak topology or with the strong Lp -topology. For this reason it is convenient to consider a space larger than W 1,p (Ω) in order to define functionals with linear growth in the gradient. This space will be the space BV (Ω) of functions with bounded variation. Working with the space BV (Ω), from (5) it follows that every minimizing sequence (uh ) of F will be bounded in BV (Ω) and converges therefore, up to a subsequence, in L1 to a BV function u (using a compactness theorem on BV (Ω)). It is then natural to extend the functional F by relaxation with respect to the L1 -topology to BV (Ω) (for the relaxed functional F we have then lower semicontinuity and coerciveness!). The main problem is then to find a representation formula for F . This problem has been studied by many mathematicians. The aim of these lectures (see Sections 5-8) is to show how to tackle with the problem of representation of F , when F has not only the volume part (i.e., is not only of the form (4)), but has also a surface energy term in its natural setting of BV functions.


85

4 Preliminaries. Functions of bounded variation and special functions of bounded variation 4.1 Notation Let us collect here briefly some notation used frequently in the following sections. If some notation is used only “locally” we will it introduce near to its use. Let n , k ∈ N be fixed. We shall denote by I an open interval of R, and by Ω a (bounded) open subset of Rn . Let us denote by A(Ω) (resp. B(Ω)) the family of the open (resp. Borel) subsets of Ω. We shall use standard notations for the Sobolev and Lebesgue spaces W 1,p (Ω; Rk ) and Lp (Ω; Rk ). When k = 1, we shall drop the target space Rk in the notation, and write just W 1,p (Ω), Lp (Ω), and so one. In particular we will denote by u p the Lp -norm of a function u in Lp (Ω). The Lebesgue measure and the (n − 1)-dimensional Hausdorff measure in Rn will be denoted by Ln and H n−1 , respectively. We shall use also the notation |E| for Ln (E), the Lebesgue measure of a measurable set E ⊂ Rn , and # for H 0 , the counting measure. Let us point out that | · | will be used also to denote the usual euclidean norm on Rn or the total variation of a measure (see below). The correct interpretation will be clear from the context or will be explained otherwise. We will denote by ·, · the usual scalar product on Rn . Let X be a set, and E ⊂ X ; we define the characteristic function of E as 1 if z ∈ E (6) 1E (z) = 0 if z ∈ X \ E, and the indicator function of E as 0 if z ∈ E χE (z) = +∞ if z ∈ X \ E.

(7)

If N ≥ 1 is an integer, then φ : RN → [ 0, +∞ ] is positively homogeneous of degree one if φ(λξ) = λφ(ξ) for every λ > 0 and for every ξ ∈ RN . Let f : RN → [ 0, +∞ ] be a convex function. We define the recession function f ∞ of f as f ∞ (ξ) = lim

t→+∞

f (tξ) t

for every ξ ∈ RN .

(8)

It is well known (see [43] that this limit exists, and that f ∞ is a Borel function, which is convex and positively homogeneous of degree one. If f (0) = 0, then f (ξ) ≤ f ∞ (ξ) for every ξ ∈ RN . Indeed, by the convexity assumption of f we get for ξ ∈ RN and t > 0 ξ ξ f (ξ) = f t + (1 − t)0) ≤ tf ( ) t t

86


and by passing to the limit as t → +∞ we get f (ξ) ≤ f ∞ (ξ). If φ : RN → [ 0, +∞ ] is a Borel function, then we denote by φ∗∗ the convex and lower semicontinuous envelope of φ, i.e., the greatest convex and lower semicontinuous function less than or equal to φ (see, for instance, [36]). If φ is finite and continuous then the following equality holds φ∗∗ (ξ) = inf

+1 !N

λi φ(ξi ) : λi ≥ 0 ,

i=1

N +1

λi = 1 ,

i=1

N +1

ξi = ξ

"

i=1

for every ξ ∈ RN . Let θ : R → [ 0, +∞[ be a subadditive function, i.e., θ(x + y) ≤ θ(x) + θ(y) for every x, y ∈ R. Assume θ(0) = 0. We can define the function θ0 : R → [ 0, +∞ ] (see, for instance, [15]) by θ0 (z) = lim

t→0+

θ(tz) θ(tz) = sup . t t t>0

(9)

The function θ0 is convex, subadditive, and positively homogeneous of degree one (it follows that θ0 (z) = zθ0 (1), where θ0 (1) is the slope (from the rightθ(tz) ≥ hand side) of θ in 0). Moreover, θ0 ≥ θ (it follows easily by θ0 (z) = sup t t>0 θ(1z)). The letters c, c , . . . , c1 , . . . will denote always a strictly positive constant, whose value may vary from line to line, and is always independent of the parameters of the problems considered each time. 4.2 Measure spaces Let us recall first the main definitions of measure theory and collect the main results we will use later. Definition 5. A function µ : B(Ω) → RN is a (vector) measure on Ω if µ(∅) = 0 and if it is countably additive; i.e., Bi Bi ∩ Bj = ∅ if i = j =⇒ µ(B) = µ(Bi ) . B= i∈N

i∈N

The set of such measures will be denoted by M (Ω; RN ). If no confusion may arise, we denote by µi (i = 1, . . . , N ) the components of µ. We say that a measure is a scalar measure if N = 1, and that it is a positive measure if it takes values in [ 0, +∞ ]. The sets of scalar and positive measures will be denoted by M (Ω) and M+ (Ω), respectively. A function µ : Bc (Ω) → RN (Bc (Ω)= family of Borel subsets of Ω with compact closure) is a Radon measure on Ω if µ|B(Ω ) is a measure on Ω for all Ω ⊂⊂ Ω. As above, we will speak of scalar and of positive Radon measures.


87

If µ ∈ M (Ω), Lp (Ω, µ; RN ) is the space of RN -valued p-summable functions with respect to µ on Ω (when µ = Ln we omit it; see above). If µ ∈ M (Ω; RN ) and B ∈ B(Ω), we define (µ B)(A) = µ(B ∩ A) for A ∈ B(Ω). We have that µ B ∈ M (Ω; RN ). If µ ∈ M (Ω; RN ) for all B ∈ B(Ω) we define the total variation of µ on B by " ! |µ(Bi )| : B = Bi , Bi pairwise disjoint . |µ|(B) = sup i∈N

i∈N

The set function |µ| is a positive finite measure on Ω (see [13], Theorem 1.6). The support of µ ∈ M (Ω; RN ) is defined as sptµ = {x ∈ Ω : |µ|(Bρ (x)) > 0 ∀Bρ (x) ⊂ Ω} . Definition 6. Let µ ∈ M+ (Ω) and λ ∈ M (Ω; RN ). We say that λ is absolutely continuous with respect to µ (and we write λ µ) if |λ|(B) = 0 for every B ∈ B(Ω) with µ(B) = 0. We say that λ is singular with respect to µ if there exists a set E ∈ B(Ω) such that µ(E) = 0 and |λ|(B) = 0 for all B ∈ B(Ω) such that B ∩ E = ∅ (we say that λ is concentrated on E). Remark 3. If f ∈ L1 (Ω, µ; RN ) and µ ∈ M (Ω) then we define the measure f µ ∈ M (Ω; RN ) by f dµ . f µ(B) = B

We have that f µ |µ|. Moreover |f µ| = |f | |µ|. Theorem 3 (Radon-Nikod´ ym). If λ ∈ M (Ω; RN ) and µ ∈ M+ (Ω), then there exists a function f ∈ L1 (Ω, µ; RN ) and a measure λs , singular with respect to µ, such that λ = f µ + λs ,

with f (x) = lim+ ρ→0

λ(Bρ (x)) for µ-a.e. x ∈ Ω. µ(Bρ (x))

This will be called the Radon-Nikod´ ym decomposition of λ with respect to µ, and the function f (given by the Besicovitch derivation theorem (see [13])) dλ . the Radon-Nikod´ ym derivative of λ with respect to µ, and denoted by dµ Remark 4. From the previous theorem we get the following results: (a) If λ µ, then λ = f µ for some f ∈ L1 (Ω, µ; RN ) ; (b) Since λ |λ| there exists ν ∈ L1 (Ω, |λ|; RN ) such that λ = ν|λ| . Since |λ| = |ν|λ| | = |ν| |λ| we get that |ν| = 1 |λ|-a.e. on Ω.

88


Let us consider the space C0 (Ω; RN ) of all continuous functions from Ω into RN which vanish on the boundary; that is, φ ∈ C0 (Ω; RN ) if ∀ε > 0 there exists a compact set Kε ⊂ Ω such that |φ| < ε on Ω \ Kε . By Riesz’s theorem we get M (Ω; RN ) = C0 (Ω; RN ) with the duality µ, φ = Ω φ dµ so that ! " |µ|(Ω) = sup φ dµ : φ ∈ C0 (Ω; RN ) , φ ∞ ≤ 1 . Ω

The usual weak∗ topology on M (Ω; RN ) is defined as the weakest topology φ dµ are continuous for every on M (Ω; RN ) for which the maps µ → φ ∈ C0 (Ω; RN ).

Ω

4.3 Functions of bounded variation We begin this section with the most common definition of BV (Ω), based on the existence of a measure distributional derivative. Definition 7. Let u ∈ L1 (Ω). We say that u is a function of bounded variation on Ω if its distributional derivative is a measure, i.e., there exists µ ∈ M (Ω; Rn ) such that ∂φ u dx = − φ dµi ∀φ ∈ Cc1 (Ω) . ∂x i Ω Ω . We denote µ = (µ1 , . . . , µn ) = Du and its components by Di u. The vector space of all functions of bounded variation in Ω is denoted by BV (Ω). Recall that Cc1 (Ω) denotes the space of C 1 functions with compact support in Ω. Note. The Sobolev space W 1,1 (Ω) ⊂ BV (Ω) ; indeed, for any u ∈ W 1,1 (Ω) the distributional derivative is given by ∇uLn , and |Du|(Ω) = Ω |∇u| dx. The inclusion is strict: there exist functions u ∈ BV (Ω) such that Du is singular with respect to Ln (for instance, the Heaviside function 1(0,+∞) , whose distributional derivative is the Dirac measure δ0 ). One of the main advantages of the BV space is that it includes, unlike Sobolev spaces, characteristic functions of sufficiently regular sets, and more generally, piecewise smooth functions. The following approximation result is useful since it often enables to restrict the proofs to smooth functions. Theorem 4. [Approximation by smooth functions] Let u ∈ BV (Ω). Then there exists a sequence (uh ) ⊂ C ∞ (Ω) such that uh → u in L1 (Ω) and |Duh |(Ω) → |Du|(Ω). Theorem 5. The following statements are equivalent: (i) u ∈ BV (Ω) ;


89

(ii) u ∈ L1 (Ω) and the total variation of u on Ω ! " V (u, Ω) = sup u divg dx : g ∈ Cc1 (Ω; Rn ) , g ∞ ≤ 1 Ω

is finite; (iii) there exists a sequence of functions (uh ) in C ∞ (Ω) such that uh → u in |∇uh | dx < +∞. L1 (Ω) and lim sup h→+∞

Ω

Note. It turns out that V (u, Ω) = |Du|(Ω) ; motivated by this fact, also |Du|(Ω) will be sometimes called the variation of u in Ω. . Note. BV (Ω) endowed with the norm u BV = u 1 + |Du|(Ω) is a Banach space. Definition 8. Let u , uh ∈ BV (Ω). We say that (uh ) weakly∗ converges in BV (Ω) to u if (uh ) converges to u in L1 (Ω) and (Duh ) weakly∗ converges to Du in Ω, i.e., φ Duh = φ Du ∀φ ∈ C0 (Ω) . lim h→+∞

Ω

Ω ∗

A simple criterion for weak convergence is stated in the following proposition. Proposition 1. Let (uh ) ⊂ BV (Ω). Then (uh ) weakly∗ converges to u in BV (Ω) if and only if (uh ) converges to u in L1 (Ω) and (uh ) is bounded in BV (Ω). Remark 5. If uh → u in L1 (Ω), then |Du|(Ω) ≤ lim inf |Duh |(Ω). h→+∞

As stated in Section 3 (Remark 2) the following compactness theorem for BV functions is very useful in connection with variational problems with linear growth in the gradient. Since the Sobolev space W 1,1 has no a similar compactness property this provides also a justification for the introduction of BV spaces in the Calculus of Variations (besides the necessity to handle in certain problems with possibly discontinuous functions). Theorem 6 (Compactness in BV ). If (uh ) ⊂ BV (Ω) and sup uh BV < h∈N

+∞, then there exists a subsequence of (uh ) converging in L1loc (Ω) to some u ∈ BV (Ω). If Ω is an open set with compact Lipschitz boundary we can say that the subsequence weakly∗ converges to u.

90


4.4 Sets of finite perimeter Definition 9. We say that a set E ⊂ Rn is a set of finite perimeter in Ω if 1E ∈ BV (Ω) ; that is, if ! " sup div g dx : g ∈ Cc1 (Ω; Rn ) , g ∞ ≤ 1 < +∞ . E

This quantity |D1E |(Ω) is called the perimeter of E in Ω. Remark 6. If E ⊂ Rn and ∂E is piecewise C 1 , then |D1E |(Ω) = H n−1 (∂E ∩ Ω). If E is of finite perimeter in Ω and C ⊂ Ω satisfies H n−1 (C) = 0, then |DχE |(C) = 0. Let E be a set of finite perimeter in Ω. The De Giorgi’s reduced boundary of E, denoted by ∂ ∗ E, is defined by " ! D1E (Bρ (x)) . = ν(x) ∈ S n−1 . ∂ ∗ E = x ∈ spt|D1E | : ∃ lim + ρ→0 |D1E (Bρ (x))| The function ν : ∂ ∗ E → S n−1 is called the interior normal to E (S n−1 denotes the unit sphere in Rn ). (Γi ) of graphs A set S ⊂ Rn is rectifiable if there exists a countable family # ∞ of Lipschitz functions of (n − 1) variables such that H n−1 (S \ i=1 Γi ) = 0. Theorem 7 (De Giorgi’s Rectifiability Theorem). Let E be a set of finite perimeter in Ω. Then (i) ∂ ∗ E is rectifiable; (ii) |D1E |(B) = H n−1 (∂ ∗ E ∩ B). In particular, H n−1 (∂ ∗ E) < +∞ ; (iii) D1E = νH n−1 ∂ ∗ E. Theorem 8 (Co-area formula for BV functions). If u ∈ BV (Ω), then the sets {u > t} are of finite perimeter for a.e. t ∈ R and |Du|(Ω) =

+∞ −∞

H n−1 (∂ ∗ {u > t} ∩ Ω) dt .

(10)

Note that this formula holds for all open sets A ⊂ Ω in place of Ω, and since both sides define finite measures, we have +∞ H n−1 (∂ ∗ {u > t} ∩ B) dt |Du|(B) = −∞

for all Borel sets B ⊂ Ω, by regularity.


91

4.5 Structure of BV functions. Approximate discontinuity points and approximate jump points Definition 10. Let u ∈ L1loc (Ω) and x ∈ Ω. We say that u has approximate limit at x if there exists z ∈ R such that 1 |u(y) − z| dy = 0 . lim ρ→0+ |Bρ (x)| Bρ (x) The set Su where this property fails is called approximate discontinuity set of u. z is uniquely determined for any point x ∈ Ω \ Su and is called the approximate limit of u at x and denoted by u %(x). If E is a set of finite perimeter, we get ∂ ∗ E ∩ Ω = S 1E ∩ Ω . If ν ∈ S n−1 , we split any ball Bρ (x) into the two halves Bρ+ (x, ν) = {y ∈ Bρ (x) : y − x, ν > 0} Bρ− (x, ν) = {y ∈ Bρ (x) : y − x, ν < 0} . Definition 11. Let u ∈ L1loc (Ω) and x ∈ Ω. We say that x is an approximate jump point at u if there exist a, b ∈ R, and ν ∈ S n−1 such that a = b and 1 |u(y) − a| dy = 0 , lim + ρ→0+ |Bρ (x, ν)| Bρ+ (x,ν) 1 lim+ − |u(y) − b| dy = 0 . ρ→0 |Bρ (x, ν)| Bρ− (x,ν) The set of approximate jump points (or jump set) of u is denoted by Ju . The triplet (a, b, ν) which turns out to be uniquely determined up to a permutation of a and b and a change of sign of ν, is denoted by (u+ (x), u− (x), %. νu (x)). On Ω \ Su we set u+ = u− = u Let u ∈ BV (Ω). By the Radon-Nikod´ ym theorem (Theorem 3) we set Du = Da u + Ds u, where Da u is the absolutely continuous part of Du with respect to the Lebesgue measure and Ds u is the singular part of Du with respect to Ln . If Dj u = Du Su is the jump part of Du and Dc u = Ds u (Ω\ Su ) is the Cantor part of u we can then write Du = Da u + Dj u + Dc u . We have that |Dc u|(B) = 0 for all Borel sets B such that H n−1 (B) < +∞. Theorem 9 (Structure of BV -functions). Let u ∈ BV (Ω). Then (i) the set Su is rectifiable and H n−1 (Su \ Ju ) = 0.

92


(ii) Moreover, Da u =

d Da u . = ∇u Ln d Ln

Dj u = (u+ − u− )νu H n−1

Ju

and νu (x) gives the approximate normal direction to Ju for H n−1 a.e. x ∈ Ju (∇u denotes the density of Da u with respect to Ln ). Finally, we have 1 |u(y) − u(x) − ∇u(x), y − x| dy = 0 lim+ |x − y| ρ→0 |Bρ (x)| Bρ (x) for a.e. x ∈ Ω (approximate differentiability of u a.e. on Ω). Note: Since for every u ∈ BV (Ω) we have H n−1 (Su \ Ju ) = 0, we will “identify” Su and Ju . 4.6 Special functions of bounded variation The space SBV (Ω) can be defined as the space of the functions u ∈ BV (Ω) such that Dc u = 0. We have then Du = ∇ Ln + (u+ − u− )νu H n−1

Su .

It turns out that SBV (Ω) contains the bounded “piecewise” Sobolev functions. More precisely, if Ω is bounded, K is a closed subset of Rn with H n−1 (K ∩Ω) < +∞ and u ∈ L∞ (Ω) with u ∈ W 1,1 (Ω \K) then u ∈ SBV (Ω) and H n−1 (Su \ K) = 0. For a thorough treatment of BV functions we refer to [13], [19]. 4.7 BV (SBV ) functions in one dimension If u ∈ BV (a, b) then it can be easily seen that there exists a constant c such that u(x) = c + Du(a, x) by differentiating both sides of this equality in the sense of distributions. Moreover, the right-hand side and the left-hand side limits 1 t+h . u(s) ds u(t+) = u+ u(t+) = lim+ h→0 h t u(t−) = lim+ h→0

1 h

t

u(s) ds

. u(t−) = u−

t−h

exist at all t ∈ [ a, b[ and t ∈ ]a, b ], respectively. Finally, u(t+) = u(t−) if |Du|({t}) = 0.


93

If M (I) is the space of all Radon measures on I with bounded total variation and given t0 ∈ [ a, b[ and u0 ∈ R (see above) there is a 1−1 correspondence between M (I) and the subspace {u ∈ BV (I) : u(t0 +) = u0 } given by µ → uµ , where u0 + µ( ]t0 , t ]) if t ≥ t0 uµ (t) = u0 − µ( ]t, t0 ]) otherwise . In treating functions in BV (I) (for instance, in Section 5) we define sometimes functions in BV (I) by simply describing their measure derivative and the value u(t0 +) at some point t0 ∈ [ a, b[ (or equivalently u(t0 −)) at some point t0 ∈ ]a, b ]. The space SBV (I) reduces to the set of functions u in BV (I) such that their measure first derivative, denoted in this case by u, ˙ is of the form u˙ = u˙ a dt + u˙ s = u˙ a dt +

∞

ak δtk ,

k=1

where tk ∈ I, ak ∈ R,

∞

|ak | < +∞, and δt is the Dirac measure at t. We

k=1

get ak = u(tk +) − u(tk −). Moreover, we will often express integration on Su as a summation: for example |u+ (t) − u− (t)| dH 0 (t) = |u(t+) − u(t−)| d#(t) = Su ∩Ω

Su ∩Ω

=

|u(t+) − u(t−)| .

t∈Su ∩Ω

5 Lower semicontinuity and relaxation in one dimension 5.1 A characterization of l.s.c. for functionals defined on SBV (I). Relaxation on BV (I) Lower semicontinuity conditions for general functionals defined on SBV take a complex form, due to the possible interaction of the Lebesgue part and the jump part. In this section we state a simplified version of a more general lower semicontinuity theorem in dimension one (see [19], Thm. 2.6). Theorem 10 (Characterization of l.s.c. for functionals defined on SBV (I)). Let f : R → [ 0, +∞[ be convex and f (z) = +∞ |z|→+∞ |z| lim

(i.e., f is superlinear at ∞).

(11)

94


Let θ : R × R → [ 0, +∞[ satisfy inf θ > 0. Let I ⊂ R be a bounded open interval. Then the functional F : BV (Ω) → [ 0, +∞ ] ⎧ ⎨ f (u˙ a ) dt + θ(u(t+), u(t−)) if u ∈ SBV (I), (12) F (u) = I t∈Su ⎩ +∞ if u ∈ BV (I) \ SBV (I) is lower semicontinuous with respect to the BV − w∗ convergence in BV (I) if and only if θ is lower semicontinuous and θ is subadditive, i.e., θ(x, y) ≤ θ(x, z) + θ(z, y)

∀x, y, z ∈ R .

(13)

Remark 7. Condition (11) can be substituted by the following one: there exists φ(z) = +∞ and f (z) ≥ φ(z) a convex function φ : R → [ 0, +∞[ with lim |z|→+∞ |z| for every z ∈ R. Remark 8. By the superlinear assumption (11) and the convexity of f we get easily that f (z) ≥ |z| − c (14) for some suitable constant c ≥ 0. If θ(x, y) ≥ α|x − y| for α > 0, x, y ∈ R, then the above lower semicontinuity result in BV − w∗ is equivalent to lower semicontinuity of F with respect to the L1 -convergence in BV (I). Indeed, under the above assumptions we get that F (u) ≥ |u|(I) ˙ − c , for some suitable constant c (by Theorem 6, this estimate implies that L1 convergent sequences converge, up to a subsequence, weakly* in BV (I)). Remark 9. Let us consider the special case of θ when θ(x, y) = ϕ(x − y) with ϕ : R → [ 0, +∞[ . Then θ is subadditive, i.e., satisfies (13) if and only if ϕ(x + y) ≤ ϕ(x) + ϕ(y)

∀x, y ∈ R .

(15)

If ϕ satisfies (15) we will again say that it is subadditive. Examples: (i) Let ϕ : [ 0, +∞[ → [ 0, +∞[ be concave. Then it is subadditive. (ii) It can be easily shown that the functions | sin z|, arctan |z|, min{|z|, 1} define subadditive functions. (iii) Let ϕ : [ 0, +∞[ → [ 0, +∞[ be concave and non-decreasing. Then φ(z) = ϕ(|z|) is subadditive on R.


95

Guided by the semicontinuity result above we can now state a relaxation result on SBV (I) (see [21], Prop. 5.4). We shall focus our attention on the behaviour of the jump-part energy. Theorem 11 (Relaxation in SBV (I) - no interaction of bulk and jump-part energy densities). Let f : R → [ 0, +∞[ be a convex function, f (0) = 0 and ∀z ∈ R . (16) |z|2 − c ≤ f (z) ≤ c (1 + |z|2 ) Let θ : R × R → [ 0, +∞[ be a Borel function satisfying infθ > 0 ,

θ(x, y) ≥ |x − y| .

(17)

Let F : BV (I) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ f (u˙ a ) dt + θ(u(t+), u(t−)) if u ∈ SBV (I), #(Su ) < +∞, F (u) = I t∈S u ⎩ +∞ otherwise on BV (I). (18) The relaxed functional with respect to the L1 (I)-topology, denoted by F , can then be written as ⎧ ⎨ f (u˙ a ) dt + sub θ(u(t+), u(t−)) if u ∈ SBV (I), #(Su ) < +∞, F (u) = I t∈S u ⎩ +∞ otherwise on BV (I), (19) where sub θ is the greatest lower semicontinuous and subadditive function less than or equal to θ (see (20), (21) and Proposition 3 below). Note. We can substitute |z|2 in the growth condition f (z) ≥ |z|2 − c with any convex function growing more than linearly at infinity, with minor modifications in the proof. Remark 10. Let F denote the relaxed functional of F with respect to the L1 topology on BV (I). Then F (u) < +∞ only for functions u ∈ SBV (I) with #(Su ) < +∞. Indeed, let u ∈ BV (I) such that F (u) < +∞. By definition of relaxation, there exists a sequence (uh ) ⊂ BV (I) such that (uh ) converges to u in L1 and F (u) = lim F (uh ) < +∞. By our assumptions on f and θ we h→+∞

get uh ∈ SBV (I) , #(Suh ) ≤ c , uh BV ≤ c , (u˙h )a L2 ≤ c. The compactness and the lower semicontinuity theorem on SBV (I) (see Proposition 2) ensures that u ∈ SBV (I) and #(Su ) < +∞. Remark 11. By our assumptions on f and θ (note f has superlinear growth at infinity, and inf θ > 0) the relaxation process does not create interaction between the bulk energy and the jump-part energy density. Furthermore, by (16) and (17) we have F (u) ≥ |u|(I) ˙ − c : hence it is equivalent to consider sequences converging with respect to the L1 (I)-topology or with respect to the BV − w∗ topology.

96


Remark 12. A generalization to higher dimension of this result (under some further technical assumptions) can be found in [20], Prop. 7.1. Before we start with the proof of Theorem 11 we state a compactness and lower semicontinuity result on SBV (I). Moreover we give some attention to the notion of sub θ and state some of its properties (useful for the proof of Theorem 11). Their proofs and some examples will be given in the Appendix A. Proposition 2. (Compactness and lower semicontinuity theorem in SBV (I)) Let (uh ) be a sequence in SBV (I) such that uh BV ≤ c ,

(u˙ h )a L2 ≤ c ,

#(Suh ) ≤ c .

Then (possibly passing to a subsequence) there exists u ∈ SBV (I) such that uh u in BV − w∗ ,

and

(u˙ h )a u˙ a in L2 (I) .

Moreover, #(Su ) ≤ lim inf #(Suh ). Furthermore, if θ : R × R → R is lower h→+∞

semicontinuous and subadditive (i.e., θ satisfies 13), then θ(u(t+), u(t−)) ≤ lim inf θ(uh (t+), uh (t−)) . h→+∞

t∈Su

t∈Suh

A proof of this result can be found in [21], Prop. 5.1. A more general SBV compactness theorem can be found in [19] (see also [8] and references therein, and [3]). Definition: The subadditive envelope sub φ of φ : R × R → R is the greatest subadditive function less than or equal to φ. It is easy to check that sub φ ∈ R ∪ {−∞} is given by the formula sub φ(x, y) = inf

m !

" φ(xk , xk−1 ) : x0 = y, xm = x, m = 1, 2, . . . .

(20)

k=1

Definition: Given a function φ : R × R → [ 0, +∞[ we define the function sub φ : R × R → [ 0, +∞[ by setting sub φ(x, y) = sub(sc− φ)(x, y) , where sc− φ is the lower semicontinuous envelope of φ (see Definition 4). Proposition 3. Let φ : R × R → [ 0, +∞[ be a function such that inf φ > 0 , Then

φ(x, y) ≥ |x − y|

∀x, y ∈ R .

(21)


97

(i) inf(sub φ) > 0 and sub φ(x, y) ≥ |x − y| ∀x, y ∈ R ; (ii) sub φ is the greatest lower semicontinuous and subadditive function less than or equal to φ. Proof of Theorem 11. Without loss of generality let us fix here I = ]0, 1[ . Let H : BV (I) → [ 0, +∞ ] be the functional on the right-hand side of (19), i.e., ⎧ ⎨ f (u˙ a ) dt + sub θ(u(t+), u(t−)) if u ∈ SBV (I), #(Su ) < +∞, H(u) = I t∈S u ⎩ +∞ otherwise on BV (I). We have to show that F (u)= (relaxed functional of F with respect to the L1 -topology)= H(u) for every u ∈ BV (I). Step 1. H(u) ≤ F (u) for every u ∈ BV (I). Since we have H(u) ≤ F (u) for all u ∈ BV (I), it is enough to show that H is L1 -lower semicontinuous on BV (I). Let u ∈ BV (I) and let (uh ) converge to u in L1 (I) such that lim H(uh ) < +∞. Hence we can suppose (uh ) ⊂ SBV (I). Since f (z) ≥ h→+∞ 2

|z| − c, and sub θ(x, y) ≥ |x − y| we get then (u˙ h )a L2 ≤ c

uh BV ≤ c .

and

Let us remark that inf sub θ > 0, and hence we have also #(Suh ∩ I) ≤

H(uh ) ≤ c < +∞ . inf sub θ

By applying Proposition 2 we get u ∈ SBV (I), #(Su ∩ I) < +∞ and sub θ(u(t+), u(t−)) ≤ lim inf sub θ(uh (t+), uh (t−)) . h→+∞

t∈Su

(22)

t∈Suh

˙ a in L2 (I). Our assumptions on f Furthermore we can suppose (u˙ h )a (u) then imply (see [33], Ex. 1.22, Ex. 1.23) f (u˙ a ) dt ≤ lim inf f ((u˙ h )a ) dt . (23) I

h→+∞

I

By (22) and (23) we get H(u) ≤ lim H(uh ). h→+∞

Step 2. H(u) ≥ F (u) for every u ∈ BV (I). By Remark 10 it is enough to prove the above inequality for u ∈ SBV (I) with #(Su ) < +∞. We have now to show that for every u ∈ SBV (I) with #(Su ) < +∞ we can build up a recovery sequence (uh ) such that uh → u and

in L1 (I)

98


H(u) ≥ lim inf F (uh ) . h→+∞

We can study the case of a single jump without losing in generality. We can suppose u ∈ SBV (I) and u˙ s = (u(t0 +) − u(t0 −))δt0 for some t0 ∈ I. By the definition of sub θ, for every h ∈ N there exist real numbers ah0 , ah1 , . . . , ahNh and bh0 , bh1 , . . . , bhNh such that we have |bh0 − u(t0 −)| ≤

1 , h2

|ahNh − u(t0 +)| ≤

1 , h2

|bhj − ahj−1 | ≤

1 h2

(24)

for every j = 1, . . . , Nh and sub θ(u(t0 +), u(t0 −)) +

Nh 1 ≥ θ(ahj , bhj ) . h j=0

(25)

By our assumptions (17) on θ, the inequality (25) implies that Nh ≤ c and

Nh

|ahj − bhj | ≤ c .

j=0

We can suppose that Nh = N independent of h ∈ N. Let us fix M ∈ N such that & 1 1 ' , t0 + ⊂ I. t0 − M M N 1 > ) we define uh ∈ SBV (I) by setting For every h ≥ M N (so that M h uh (0+) = u(0+)

and u˙ h = wh (t) dt +

N

(ahj − bhj )δ(t0 − N −j ) , h

j=0

where

⎧ 1 ⎪ if t < t0 − ⎪ u˙ a (t) ⎪ M ⎪ ⎪ ⎪ 1 N ⎪ Mh h ⎪ u˙ a (t) + h−M ⎪ N (b0 − u(t0 −) if t0 − M ≤ t < t0 − h ⎪ ⎪ ⎪ ⎪ N −j+1 ⎪ ⎨ (bh − ah )h ≤ t and if t0 − j j−1 h wh (t) = N −j ⎪ ⎪ ⎪ t < t0 − 1≤j≤N ⎪ ⎪ h ⎪ ⎪ 1 ⎪ ⎪ ⎪ u˙ a (t) + M (u(t0 +) − ahN ) if t0 ≤ t < t0 + ⎪ ⎪ M ⎪ ⎪ 1 ⎩ if t ≥ t0 + u˙ a (t) M


(obviously wh ∈ L1 (I \ Suh ) and |(u˙ h )s | =

N

99

|ahj − bhj | < +∞). Remark that

j=0 1 1 , t0 + M [ for every h. Moreover, t0 − N uh (t) = u(t) outside ]t0 − M h → t0 as 1 1 , t0 + M [. We then have h tends to +∞ and (wh ) converges to u˙ a on ]t0 − M

uh → u and

in L1 (I)

F (uh ) =

1 1 I\[t0 − M ,t0 + M ]

+

f ((u˙ h )a ) dt +

= 1 1 I\[t0 − M ,t0 + M ]

f ((u) ˙ a ) dt +

1 1 ]t0 − M ,t0 + M [

f (wh ) dt +

N

θ(ahj , bhj )

j=0

≤

f ((u˙ h )a ) dt+

θ(uh (t+), uh (t−))

t∈Suh

1 1 ]t0 − M ,t0 + M [

1 1 I\[t0 − M ,t0 + M ]

f ((u) ˙ a ) dt +

1 1 ]t0 − M ,t0 + M [

+sub θ(u(t0 +), u(t0 −)) +

1 h

f (wh ) dt+

. (26)

Now

1 1 ]t0 − M ,t0 + M [

f (wh ) dt =

1 t0 − M

1 t0 + M

+

t0 − N h

f (u˙ a + ε(h)) dt +

t0

f (ε (h)) dt+

t0 − N h

f (u˙ a + ε (h)) dt ,

t0

where ε(h), ε (h) and ε (h) are functions which vanish as h tends to +∞ (recall (24)); by continuity we get f (wh ) dt = f (u˙ a ) dt . lim h→+∞

1 1 ]t0 − M ,t0 + M [

1 1 ]t0 − M ,t0 + M [

Hence, by passing to the limit in (26) we get lim sup F (uh ) ≤ f (u˙ a ) dt + sub θ(u(t0 +), u(t0 −)) = H(u) . h→+∞

I

It is clear that in the same way we can treat the general case of more then one jump. Remark 13. Let us note that in the previous proof Step 1 (that is H ≤ F ) has been proved using a lower semicontinuity result in SBV (I) (see Proposition

100


2) whereas Step 2 (that is H ≥ F ) has been shown building up a “recovering” sequence. As we will see, this scheme will be used more in general as far as the first step is concerned, while for Step 2 we have in general (see Section 7 and comments in Section 8) to proceed in a completely different, more technical manner. 5.2 A sufficient condition for l.s.c. for functionals defined on SBV (I). Relaxation in BV (I) Let us state the following semicontinuity theorem for a functional F like (12) where the assumption on θ to satisfy inf θ > 0 is dropped, but a condition of superlinearity at the origin is added (see [5]; see also [19]). Subsequently we will prove a relaxation result when possibly inf θ = 0 and no superlinearity condition at the origin is assumed. Theorem 12 (Lower semicontinuity - superlinear growth conditions for the bulk and jump-part energy densities). Let f : R → [0, +∞] be convex and f (z) = +∞ |z|→+∞ |z| lim

(i.e., f is superlinear at ∞).

(27)

Let θ : R → [0, +∞] be lower semicontinuous and subadditive (see (15) and θ(z) = +∞ |z|→0 |z| lim

(i.e., θ is superlinear at 0).

(28)

Let I ⊂ R be a bounded open interval. Then the functional F : BV (I) → [0, +∞] defined by ⎧ ⎨ f (u˙ a ) dt + θ(u(t+) − u(t−)) if u ∈ SBV (I), F (u) = I t∈Su ⎩ +∞ if u ∈ BV (I) \ SBV (I) (29) is BV − w∗ lower semicontinuous on BV (I). Remark 14. Let us note that the hypotheses on f and θ imply that f (z) ≥ |z| − c, and θ(z) ≥ |z| − c for some suitable non-negative constants c and c . Since in general c = 0, we get F (u) ≥ |u|(I) ˙ − c #(Su ∩ I) − c .

(30)

For functionals satisfying (30) it is in general not equivalent to consider sequences converging with respect to the L1 -topology or with respect to the BV − w∗ topology (u ∈ BV (I) or u ∈ SBV (I) does not imply #(Su ∩ I) < +∞!).


101

Remark 15. We can take for example θ(z) = |z|, z ∈ R ; more in general, every function θ(z) = ϕ(|z|), z ∈ R with ϕ : [ 0, +∞[ → [ 0, +∞[ concave, nonϕ(z) = +∞ satisfies the hypotheses of Theorem 12 (see decreasing and lim+ z z→0 Remark 9 Example (iii)). In particular, θ(z) = c ∈ R can be considered. In the higher dimensional case this result can be found in [5]. A lower semicontinuity result with respect to the L1 -topology for functionals of the form (29) with θ = θ(t, z) = 1 + a(t)|z|, and a : I → R strictly positive and continuous, can be found in [30]. Note: The function θ(x, y) = |x − y| does not fit the assumptions of Theorem 10 (indeed inf θ = 0) as well as the function θ(z) = |z| does not satisfies hypothesis (28) of Theorem 12. The following theorem deals then with the “linear” case, and no superlinearity conditions for f are required. Theorem 13 (Relaxation in BV (I) - “linear” growth case: interaction between bulk and jump-part energy densities). Let f : R → [ 0, +∞[ be convex such that f (0) = 0, and let I ⊂ R be a bounded open interval. Let F : BV (I) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ f (u˙ a ) dt + |u(t+) − u(t−)| if u ∈ SBV (I), (31) F (u) = I t∈S u ⎩ +∞ otherwise on BV (I). The relaxed functional with respect to the L1 (I)-topology, denoted by F , can be written as ˙s ∞ u |u˙ s | F (u) = f (u˙ a ) dt + f u ∈ BV (I) , (32) |u˙ s | I I ∞

where f (z) = (f (z)∧|z|)∗∗ for every z ∈ R and f (z) is its recession function. Remark 16. Note that the right-hand side of (32) is defined on the whole u˙ s denotes the Radon-Nikod´ ym derivative of u˙ s with reBV (I). Recall that |u˙ s | spect to |u˙ s | and that we use the notation

h(x) |u˙ s | instead of I

h(x) d|u˙ s |. I

Remark 17. Let us point out that the energy densities which appear in the integral representation (32) take into account simultaneously both the bulk energy density f (z) as well as the jump-part energy density θ(z) = |z|. If we take now f and θ as in Theorem 13 we have immediately θ0 (z) = θ(z) (see (9). Moreover (see (83)) f (z) = (f (z) ∧ θ0 (z))∗∗

102


and

∞

f (z) = sub (f ∞ (z) ∧ θ(z)) .

We will notice briefly the competition between the functions f and θ in order to find a “right energy balance”: the function f will be preferred with respect to the function θ if θ has a fast growth for small z (that is, regularity will be preferred instead of small jumps); on the other hand, if the function f grows more than linearly at infinity the function θ should be selected (that is, fast growth of the functions will be replaced by jumps). Example 1. Take f (z) = z 2 ; we get easily that 2 z if |z| ≤ 12 f (z) = 1 |z| − 4 otherwise, and ∞

f (z) = lim

t→+∞

f (tz) = |z| . t

In order to prove Theorem 13 we shall need the following results (stated here for our convenience in dimension 1) about relaxation in BV and W 1,1 , due to Goffman and Serrin, and Buttazzo and Dal Maso, respectively. Theorem 14. ([39]) Let V : R → [ 0, +∞[ be a convex function such that 0 ≤ V (z) ≤ c(1 + |z|) . Let E : BV (I) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ V (u˙ a ) dt if u ∈ W 1,1 (I), E(u) = I ⎩ +∞ if u ∈ BV (I) \ W 1,1 (I). Then the relaxed functional of E with respect to the L1 -topology is given by u˙ s |u˙ s | E(u) = V (u˙ a ) dt + V ∞ (33) |u˙ s | I I for every u ∈ BV (I). Theorem 15. ([28]) Let V : R → [ 0, +∞[ be a Borel function such that c1 |z| − c2 ≤ V (z) ≤ c(1 + |z|) . Let E : W 1,1 (I) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ V (u˙ a ) dt if u ∈ C 1 (I) ∩ W 1,1 (I), E(u) = I ⎩ +∞ otherwise on W 1,1 (I).


103

Then the relaxed functional of E with respect to the L1 -topology is given by E(u) = V ∗∗ (u˙ a ) dt (34) I

for every u ∈ W

1,1

(I).

Proof of Theorem 13 (see [30], Theorem 3.1, and [21], Section 4). Let H : BV (I) → [ 0, +∞ ] be the functional on the right-hand side of (32), i.e., ˙s ∞ u |u˙ s | . H(u) = f (u˙ a ) dt + f | u ˙ s| I I We have to show that F (u)= (relaxed functional of F with respect to the L1 -topology)=H(u) for every u ∈ BV (I). Step 1. H(u) ≤ F (u) for every u ∈ BV (I). Let us note that f (z) = (f (z) ∧ |z|)∗∗ ≤ |z| for all z ∈ R. Hence we can apply Theorem 14 with V (z) = f (z) ; hence H(u) is lower semicontinuous with respect to the L1 -topology on BV (I) (by definition of relaxation). Now it is enough to show that H(u) ≤ F (u) for every u ∈ BV (I): clearly f (z) = (f (z) ∧ |z|)∗∗ ≤ f (z)

∀z ∈ R .

f (tz) |tz| ≤ lim = |z| and hence t→+∞ t t u˙ s ˙s ∞ u |u˙ s | ≤ |u˙ s | = |u˙ s | = |u˙ s |(I) . f |u˙ s | ˙ s| I I |u I ∞

Furthermore, f (z) = lim

t→+∞

Since |u˙ s |(I) ≤

⎧ ⎨ |u(t+) − u(t−)| if u ∈ SBV (I), ⎩ t∈Su +∞

on BV (I) \ SBV (I)

we may conclude that H(u) ≤ F (u) for every u ∈ BV (I). By definition of relaxation we get then H ≤ F on BV (I). Step 2. H(u) ≥ F (u) for every u ∈ BV (I). It is enough to show that ⎧ ⎨ f%(u˙ a ) dt if u ∈ C 1 (I) ∩ W 1,1 (I), (35) F (u) ≤ I ⎩ +∞ otherwise on W 1,1 (I), where f%(z) = f (z) ∧ |z|. Indeed, by Theorem 15 with V (z) = f%(z) (recall that by our assumptions we have c|z| − c ≤ f (z) for suitable positive constants c and c ) we get F (u) ≤ f (u˙ a ) dt for every u ∈ W 1,1 (I). (36) I

104


Since clearly F (u) ≤

⎧ ⎨

f (u˙ a ) dt if u ∈ W 1,1 (I),

⎩ +∞ I

if u ∈ BV (I) \ W 1,1 (I),

by applying again Theorem 14 with V (z) = f (z) we get finally F (u) ≤ H(u)

∀u ∈ BV (I) .

To conclude the proof of Step 2 it remains therefore to prove (35). Let u ∈ C 1 (I) ∩ W 1,1 (I) : let us set Iu = {t ∈ I : f (u˙ a (t)) > |u˙ a (t)|} (recall that u˙ a coincides with the classical derivative of u in C 1 ). We approximate u on Iu with piecewise constant functions, thus replacing quick growth of u by a sequence of jumps. The set Iu is open, hence, considering its connected components, it can be split in an at most countable union of open intervals of I: ∞ Iui . Iu = i=1

We construct a sequence (uk )k as follows: fixed h ∈ N we subdivide every interval Iui in a finite number of contiguous subintervals of length less than 1/h : ni,h 1 i,h i,h Im |Im |< . Iui = h m=1 i,h i,h i Set ai,h m = inf Im for m ≤ ni,h and ani,h = sup Iu . Let us consider now for #k every k ∈ N the set i=1 Iui and let us set

uhk (t) =

⎧ k ni,h k ⎪ ⎪ i,h ⎪ u(a )1 (t) if t ∈ Iui , i,h ⎪ m Im ⎨ i=1 m=1

⎪ ⎪ ⎪ ⎪ ⎩ u(t)

i=1

if t ∈ I \

k

Iui .

i=1

Let us take now h = h(k) such that h(k) → +∞ as k → +∞ and uhk − u L1 (I)
1 , α , β > 0 and α ≤ θ(x, y) ≤ β(1 + |x − y|) for all x, y ∈ R. Let us now present a relaxation result for functionals which cannot be handled by Theorem 16 since in particular a condition like (40) fails. This result is the n-dimensional version of Theorem 13. In this case, the lower bound for the relaxed functional will be obtained as in the 1-dimensional case while for the upper bound we must use fine properties of the BV functions (in particular, we will use the co-area formula for BV functions to construct the “recovery”-sequence (see [21], Theorem 3.1). Theorem 17 (Relaxation in BV (Ω) - “linear” growth case: interaction between bulk and surface energy densities). Let f : Rn → [ 0, +∞ ] be a lower semicontinuous and convex function, and f (0) = 0. Furthermore, assume that the set (42) K = {z ∈ Rn : f (z) ≤ |z|} is bounded. Let Ω be a bounded open subset of Rn . Let F : BV (Ω) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ f (∇u) dx + |u+ − u− | dH n−1 if u ∈ SBV (Ω), F (u) = Su ∩Ω ⎩ Ω +∞ otherwise on BV (Ω). (43) The relaxed functional with respect to the L1 (Ω)-topology, denoted by F , can be written for u ∈ BV (Ω) as ∞ Ds u ) |Ds u| F (u) = f (∇u) dx + f ( |Ds u| Ω Ω (44) f (∇u) dx + |Ds u| , = Ω

Ω

where f (z) = (f (z) ∧ |z|)∗∗ for every z ∈ Rn and f

∞

is its recession function.


109

Ds u is the Radon-Nikod´ ym derivative of the measure |Ds u| h(x) |Ds u| instead Ds u with respect to its total variation, and that we write Ω of h(x) d|Ds u|. Remember that

Ω

Remark 20. The above theorem can be extended to the vector-valued case under the same hypotheses on f . Moreover, this result can be obtained easily as a corollary of a more general relaxation result proved by Braides and Coscia ([22]) and treated in Section 7 (see Corollary 2). Remark 21. Let us remark that, once f = +∞, it is not restrictive to suppose that f (0) = 0. In fact, otherwise we could consider the function f1 (z) = f (z + z0 ) − f (z0 ), where f (z0 ) = min f . ∞ Notice also that we have f (z) = |z| : indeed, it is easy to see that there exists R > 0 such that K ⊂ BR (0)

and

|z| − R ≤ f (z) ≤ |z| .

(45)

f (tz) ∞ = |z|. Since f (z) = t Ds u |z|, we get immediately the last equality in (44) since = 1 |Ds u|−a.e. |Ds u| on Ω. ∞

By (45) we get then immediately f (z) = lim

t→+∞

Remark 22. Let us underline that assumption (42) implies that the L1 (Ω)relaxed functional of F coincides with the BV − w∗ -relaxed functional of F (if Ω has a compact Lipschitz boundary). Indeed, let us denote by F BV −w∗ the relaxed functional of F with respect to the BV − w∗ topology. Clearly we have F BV −w∗ ≥ F . On the other hand, let (uh ) ⊂ BV (Ω) such that (uh ) converges to u in L1 (Ω) and F (u) = lim F (uh ) < +∞. For every h ∈ N h→+∞

consider Ωh = {x ∈ Ω : ∇uh (x) ∈ K} = {x ∈ Ω : f (∇uh (x)) ≤ |∇uh (x)|} and set Ωh = Ω \ Ωh . We get then − n−1 |Duh | = |∇uh (x)| dx + |∇uh (x)| dx + |u+ h − uh | dH Ωh Ω S ∩Ω u h − n−1 ≤ R|Ωh | + f (∇uh (x)) dx + |u+ h − uh | dH ≤ R|Ω| + c .

Ωh

Su ∩Ω

(Remember that for every x ∈ Ωh we have |∇uh (x)| ≤ R ; moreover, we have supposed that limh F (uh ) exists finite). Passing possibly to a subsequence we obtain that uh u BV − w∗ and therefore F BV −w∗ (u) ≤ lim inf F (uh ) = h

F (u).

110


Let us state now in dimension n > 1 the relaxation theorems by Goffman and Serrin and by Buttazzo and Dal Maso, respectively, used in Section 5 for n = 1. Theorem 18. ([39]) Let V : Rn → [ 0, +∞[ be a convex function such that 0 ≤ V (z) ≤ c(1 + |z|) . Let E : BV (Ω) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ V (∇u) dx if u ∈ W 1,1 (Ω), E(u) = Ω ⎩ +∞ if u ∈ BV (Ω) \ W 1,1 (Ω). Then the relaxation of E with respect to the L1 -topology is given for any u ∈ BV (Ω) by Ds u ) |Ds u| . E(u) = V (∇u) dx + V ∞( (46) |Ds u| Ω Ω Theorem 19. ([28]) Let V : Rn → [ 0, +∞[ be a Borel function such that c1 |z| − c2 ≤ V (z) ≤ c(1 + |z|) . Let E : W 1,1 (Ω) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ V (∇u) dx if u ∈ C 1 (Ω) ∩ W 1,1 (Ω), E(u) = Ω ⎩ +∞ if u ∈ W 1,1 (Ω) \ C 1 (Ω). Then the relaxed functional of E with respect to the L1 -topology is given by E(u) = V ∗∗ (∇u) dx (47) Ω

for every u ∈ W 1,1 (Ω). Proof of Theorem 17. Let H : BV (Ω) → [ 0, +∞ ] be the functional on the right-hand side of (44), i.e., ∞ Ds u ) |Ds u| . H(u) = f (∇u) dx + f ( |Ds u| Ω Ω We have to show that F (u)= (relaxed functional of F with respect to the L1 (Ω)-topology)= H(u) for every u ∈ BV (Ω). Step 1. H(u) ≤ F (u) for every u ∈ BV (Ω). Let us note that f (z) = (f (z) ∧ |z|)∗∗ ≤ |z| ∀z ∈ Rn . We can apply Theorem 18 with V (z) = f (z) ; hence


111

H(u) is lower semicontinuous with respect to the L1 (Ω)-topology on BV (Ω). ∞ Since f (z) ≤ |z| ∀z ∈ Rn and f (z) = |z| we have immediately that ⎧ ⎨ |u+ − u− | dH n−1 if u ∈ SBV (Ω), |Ds u|(Ω) ≤ Su ∩Ω ⎩ +∞ on BV (Ω) \ SBV (Ω) and hence H(u) ≤ F (u) for every u ∈ BV (Ω). By definition of relaxation we get then H ≤ F on BV (Ω). Step 2. H(u) ≥ F (u) for every u ∈ BV (Ω). It is enough to show that ⎧ ⎨ f%(∇u) dx if u ∈ C 1 (Ω) ∩ W 1,1 (Ω), (48) F (u) ≤ Ω ⎩ +∞ otherwise on W 1,1 (Ω), where f%(z) = f (z) ∧ |z| =

f (z) if z ∈ K, |z| otherwise.

Indeed, by Theorem 19 [28] we get then (by taking V (z) = f%(z)) F (u) ≤ f (∇u) dx for every u ∈ W 1,1 (Ω). Ω

Since clearly F (u) ≤

⎧ ⎨

f (∇u) dx if u ∈ W 1,1 (Ω),

⎩ +∞ Ω

otherwise on BV (Ω),

by applying Theorem 18 ([39]) (with V (z) = f (z)) we get finally F (u) ≤ H(u)

∀u ∈ BV (Ω) .

To conclude the proof of Step 2 it remains therefore to prove (48). Let us consider a function u ∈ C 1 (Ω) ∩ W 1,1 (Ω) (hence, in particular, u ∈ C 1 (Ω) ∩ BV (Ω)). Let us consider the set Ω = {x ∈ Ω : ∇u(x) ∈ K} = {x ∈ Ω : f (∇u(x)) ≤ |∇u(x)|} . Let us recall that if x ∈ Ω , then |∇u(x)| ≤ R (since K ⊂ BR (0)). For every ε > 0 we take an open set of finite perimeter Ωε ⊂ Ω such that |Ω \ Ωε | ≤ ε .1 1

it suffices to consider for example the set Ωε = {x ∈ Ω : dist(x, ∂Ω ) > η} for η = η(ε) > 0 small enough. Remember that dist(·, ∂Ω ) is a Lipschitz function and that {u > t} is a set of finite perimeter for a.e. t ∈ R whenever u is Lipschitz.

112


We construct a sequence (uh ) piecewise constant on Aε = Ω \ Ω ε as follows: by the co-area formula (10) for any fixed h ∈ N we have |Du|(Aε ) =

k∈Z

k+1 h k h

H n−1 (∂ ∗ {u > t} ∩ Aε ) dt .

By the mean value theorem there exists shk ∈ 1 n−1 ∗ H (∂ {u > shk } ∩ Aε ) ≤ h We can then define

uh =

k+1 h k h

&k k + 1' , such that h h

H n−1 (∂ ∗ {u > t} ∩ Aε ) dt .

k on {shk−1 shk } ∩ Aε ) + H n−1 ((∂ ∗ Ωε ) ∩ Ω) ≤ h h k∈Z k+1 h 1 H n−1 (∂ ∗ {u > t} ∩ Aε ) dt + H n−1 ((∂ ∗ Ωε ) ∩ Ω) ≤ k h k∈Z +∞h 1 = H n−1 (∂ ∗ {u > t} ∩ Aε ) dt + H n−1 ((∂ ∗ Ωε ) ∩ Ω) h −∞ 1 n−1 ∗ = |∇u| dx + H ((∂ Ωε ) ∩ Ω) h Aε 1 ≤ |∇u| dx + εR + H n−1 ((∂ ∗ Ωε ) ∩ Ω) < +∞ . h Ω\Ω (51) Recall that Ωε has finite perimeter, hence H n−1 ((∂ ∗ Ωε ) ∩ Ω) is finite (see Theorem 7). We can then estimate (using (51)) − n−1 F (uh ) = f (∇uh ) dx + |u+ h − uh | dH Ω S ∩Ω u h − n−1 = f (∇u) dx + |u+ h − uh | dH Ω S ∩Ω ε uh 1 ≤ f (∇u) dx + |∇u| dx + εR + H n−1 ((∂ ∗ Ωε ) ∩ Ω) . h Ω Ω\Ω

Since uh → u in L∞ (Ω) we obtain F (u) ≤ lim inf F (uh ) ≤ f (∇u) dx + h

Ω

Ω\Ω

|∇u| dx + εR


113

for any ε > 0. By the arbitrariness of ε we conclude that for every u ∈ C 1 (Ω) ∩ W 1,1 (Ω) we have F (u) ≤ f%(∇u) dx , with f%(z) = f (z) ∧ |z|, Ω

and (48) follows.

7 Lower semicontinuity and relaxation in higher dimension and for vector-valued functions Let us start with some notation and definitions not given previously. We shall denote by Mk×n the space of k × n matrices (k rows, n columns) and by the subset of Mk×n of all matrices with rank less than or equal to one. Mk×n 1 We shall identify Mk×n with Rkn . If a ∈ Rk and b ∈ Rn , the tensor product is the matrix whose entries are ai bj with i = 1, . . . , k, and a ⊗ b ∈ Mk×n 1 j = 1, . . . , n. Conversely, if a matrix ξ has rank one, there are two vectors a ∈ Rk and b ∈ Rn such that ξ = a ⊗ b. If A ⊂ Rk and b ∈ Rn , we set . Remark that |a ⊗ b| = |a||b| (the norms are A ⊗ b = {a ⊗ b : a ∈ A} ⊂ Mk×n 1 taken in the proper spaces). Definition 12. (see [40, 41, 31, 32, 23]) We say that a continuous function ϕ : Mk×n → [ 0, +∞[ is quasiconvex if |A|ϕ(ξ) ≤ ϕ(ξ + ∇u(x)) dx (52) A

for every ξ ∈ M

k×n

, A bounded open subset of Rn and u ∈ Cc1 (A; Rk ).

This property is a well-known necessary and sufficient condition for the lower semicontinuity of multiple integrals in Sobolev spaces (see [1], [31]) playing the same role of the convexity in the case of scalar functions (condition (52) is equivalent to convexity in the case k = 1). Let us recall that if we allow the function u to be vector-valued, i.e., u ∈ W 1,p (Ω; Rk ), then the convexity hypothesis turns out to be sufficient, but too strong to be necessary f (∇u) dx to be weakly lower semicontinuous on W 1,p (Ω; Rk ).

for F (u) = Ω

Remark 23. Every quasiconvex function ϕ : Mk×n → [ 0, +∞[ is rank-one convex, i.e., ϕ satisfies ϕ(λξ + (1 − λ)η) ≤ λϕ(ξ) + (1 − λ)ϕ(η) for every ξ, η ∈ Mk×n such that rank(ξ − η) ≤ 1, and every λ ∈ [0, 1] (see ˇ ak shows in [45] that the converse is not true. [31], [32], [23]). V. Sver´

114


Before we start with a lower semicontinuity result on SBV (Ω; Rk ) for functionals of the form f (∇u) dx + g(u+ − u− , νu ) dH n−1 , (53) F (u) = Ω

Su ∩Ω

let us introduce here briefly the space BV (Ω; Rk ) and SBV (Ω; Rk ). All the notions and properties (like Su , approximate limit, approximate differential, rectifiability of Su , . . .) which can be extended naturally from k = 1 to k > 1 will be omitted (see, for example, [13]). We say that u ∈ L1 (Ω; Rk ) belongs to BV (Ω; Rk ) if for any i ∈ {1, . . . , k} and j ∈ {1, . . . , n} there is a measure µji with finite total variation in Ω such that ∂φ u(i) dx = − φ dµji ∀φ ∈ Cc1 (Ω) . ∂xj Ω Ω We denote by Du the Mk×n -valued measure whose components are the µji , and by |Du| its total variation. Also for a function u ∈ BV (Ω; Rk ) we have the Lebesgue decomposition Du = Da u + Ds u = ∇uLn + Ds u , where we denote by ∇u the density of the absolutely continuous part of Du with respect to the Lebesgue measure. The singular part of Du with respect to the Lebesgue measure can be further decomposed into two mutually singular measures as Ds u = (u+ − u− ) ⊗ νu H n−1 Su + Dc u , where (u+ − u− ) ⊗ νu H n−1

Su is the Hausdorff part and Dc u the Cantor Ds u the Radon-Nikod´ ym derivative of Ds u with part of Du. We indicate by |Ds u| respect to its total variation. We say that u ∈ SBV (Ω; Rk ) if u ∈ BV (Ω; Rk ) and Dc u = 0. Theorem 20 (Lower semicontinuity for functionals on SBV (Ω; Rk )). Let f : Mk×n → [ 0, +∞[ be quasiconvex, and satisfy a growth condition of order p > 1, i.e., |z|p ≤ f (z) ≤ c(1 + |z|p )

∀z ∈ Mk×n .

(54)

∀s ∈ Rk , ∀ν ∈ Rn

(55)

Let g : Mk×n → [ 0, +∞[ be of the form g(s, ν) = θ(|s|)ψ(ν) satisfying g(s, ν) ≥ c > 0 ,

(56)

with θ : ]0, +∞[ → ]0, +∞[ concave, non decreasing, and ψ : R → [ 0, +∞[ convex, even, positively 1-homogeneous. Let Ω be an open subset of Rn . Then the functional defined in (53), i.e., n


F (u) =

f (∇u) dx + Su ∩Ω

Ω

115

g(u+ − u− , νu ) dH n−1

is L1loc (Ω; Rk )-lower semicontinuous on SBV (Ω; Rk ). The proof of this result can be found in [7]. Remark 24 (Relaxation on SBV (Ω; Rk ) - the superlinear growth case). Notice that a corresponding relaxation result has been obtained by Fonseca and Francfort in [37]: they study the relaxation with respect to the L1 (Ω; Rk )topology in SBV (Ω; Rk ) of the functional f (∇u) dx + λH n−1 (Su ∩ Ω) λ>0 (57) Ω

(note: g(s, ν) ≡ λ), where f has p-growth (p > 1) and satisfies a local Lipschitz condition. The relaxed functional related to (57) is defined only on SBV (Ω; Rk ) and has the form Qf (∇u) dx + λH n−1 (Su ∩ Ω) , Ω

where Qf is the quasiconvexification of f given by Qf (ξ) = inf{ f (ξ + ∇u(x)) dx : u ∈ Cc1 (Ω; Rk )} . Ω

Let us note that if f : M

k×n

→ R is a locally bounded Borel function, then

Qf = sup{ϕ : Mk×n → R : ϕ quasiconvex, ϕ ≤ f } is the quasiconvex envelope of f . Without assuming any local Lipschitz condition on f in (57) a completely analogous result has been proved in [20]. Remark 25 (Relaxation on SBV (Ω; Rk ) - the linear growth case: interaction between bulk and surface energy densities). In [14], Barroso, Bouchittè, Buttazzo and Fonseca proved a relaxation result for (53) (the integrand functions f and g possibly depending also on the x-variable) with respect to the L1 -convergence in the space SBV (Ω; Rk ), where they assumed that f is quasiconvex, and has linear growth, and that g(·, ν) grows also linearly, i.e., c|ξ| ≤ f (ξ) ≤ C(1 + |ξ|), and c1 |η| ≤ g(η, ν) ≤ C1 |η| (when g ≡ 0, i.e., the relaxation of quasiconvex functionals in BV (Ω; Rk ) for integrands f (x, u, ∇u) has been performed by Fonseca and M¨ uller (see [38])). This result extends “partially” the paper by [17] since they are working with linear growth assumptions on f or g, but they had to make an isotropy assumption on f and g (the result of this paper will be stated in Section 8).

116


In the remaining part of this section we will present a relaxation result in BV (Ω; Rk ) with respect to the L1 -topology of the functional f (∇u) dx + g((u+ − u− ) ⊗ νu ) dH n−1 , F (u) = Ω

Su ∩Ω

obtained by Braides and Coscia in [22] under very mild assumptions on f and under the assumption that g is positively 1-homogeneous. The main motivation to present the proof of this relaxation result lies in the desire to show an often used scheme of proof followed in this context (see Remark 13) forced by the fact that for vector-valued functions no co-area formula is available. We will state the theorem with a stricter assumption on g as that given in the original paper. In other words, instead of requiring g locally bounded on ), one gets the same relaxation result under rank-one matrices (i.e., on Mk×n 1 the hypothesis that there exist n linearly independent vectors ν1 , . . . , νn in S n−1 such that g is locally bounded on Rk ⊗ νm for all m = 1, . . . , n. Theorem 21 (Relaxation in BV (Ω; Rk ) - no growth assumptions on the bulk energy and the surface energy densities: interaction of the two energy densities by relaxation). Let f : Mk×n → [ 0, +∞ ] be a → [ 0, +∞[ be a positive Borel function such that f (0) = +∞ ; let g : Mk×n 1 . Let Ω be positively 1-homogeneous Borel function, locally bounded on Mk×n 1 a bounded open subset of Rn . Let F : BV (Ω; Rk ) → [ 0, +∞ ] be the functional defined by ⎧ ⎨ f (∇u) dx + g((u+ − u− ) ⊗ νu ) dH n−1 if u ∈ SBV (Ω; Rk ) F (u) = Ω Su ∩Ω ⎩ +∞ otherwise. (58) Then the relaxed functional with respect to the L1 (Ω; Rk )-topology is given by Ds u |Ds u| F (u) = ϕ(∇u) dx + ϕ∞ (59) |Ds u| Ω Ω for every u ∈ BV (Ω; Rk ), where the function ϕ : Mk×n → [ 0, +∞[ is given by } (60) ϕ(ξ) = sup{ψ(ξ) : ψ quasiconvex, ψ ≤ f on Mk×n , ψ ∞ ≤ g on Mk×n 1 and satisfies 0 ≤ ϕ(ξ) ≤ c(1 + |ξ|) for every ξ ∈ Mk×n . Remark 26. If we suppose in addition that f is convex, and g is rank-one convex on Rk ⊗ ν for every ν ∈ S n−1 (i.e., g(λa ⊗ ν + (1 − λ)b ⊗ ν) ≤ λg(a ⊗ ν) + (1 − λ)g(b ⊗ ν) , ∀a, b ∈ Rk , λ ∈ [0, 1] , ν ∈ S n−1 ), then F is convex on SBV (Ω; Rk ), and we get that also the relaxed functional F must be convex, and hence also the integrand ϕ. Then we have the formula }. ϕ(ξ) = sup{ψ(ξ) : ψ convex, ψ ≤ f on Mk×n , ψ ∞ ≤ g on Mk×n 1


117

In the scalar case (k = 1) we get as a corollary of Theorem 21 a generalization of the relaxation theorem 17, where ϕ(z) = (f (z) ∧ |z|)∗∗ (in this case we supposed f (0) = 0.) Corollary 2. Let k = 1. Under the hypotheses of Theorem 21, ϕ satisfies the formula ∀z ∈ Rn . ϕ(z) = (f ∧ (f (0) + g))∗∗ (z) Proof. Let ψ be a convex function such that ψ ≤ f and ψ ∞ ≤ g on Rn . In particular ψ(0) ≤ f (0), and we get then that ψ ≤ f (0) + g .

(61)

Indeed, for 0 < t ≤ 1, z ∈ Rn , using the convexity of ψ z z z ψ(z) = ψ(t ) ≤ tψ( ) + ψ(0) ≤ tψ( ) + f (0) . t t t By passing to the limit as t → 0+ , we get ψ(z) ≤ ψ ∞ (z) + f (0) ≤ g(z) + f (0) proving (61). By Remark 26 we conclude that ϕ(z) = sup{ψ(z) : ψ convex, ψ ≤ f ∧ (f (0) + g)} = (f ∧ (f (0) + g))∗∗ (z) as desired. Remark 27. Let H : BV (Ω; Rk ) → [ 0, +∞[ be the functional on the righthand side of (59), i.e., Ds u |Ds u| ϕ(∇u) dx + ϕ∞ H(u) = |Ds u| Ω Su ∩Ω with ϕ given by (60). We have to show that F (u)=(relaxed functional of F with respect to the L1 (Ω; Rk )-topology)= H(u) for every u ∈ BV (Ω; Rk ). Analogously to the other proofs of relaxation we will show that H ≤ F and H ≥ F . While the proof of the lower bound (i.e., H ≤ F ) is completely analogous to the previous ones (based on a semicontinuity result for H and the inequality H ≤ F which can be proved using the definition of ϕ in (60)), the proof of the upper bound (i.e., F ≤ H) will be completely different and we have to work hard: 7.1: we will first establish that under the very weak hypotheses on f and g the relaxed functional F has linear growth, and hence, is finite on the whole BV (Ω; Rk ) ; 7.2: then we will prove by a measure theoretical approach and a localization technique that the study of the relaxed functional at a fixed u can be reduced to the study of a regular Borel measure on Ω ;

118


7.3: this fact allows us to use some integral representation arguments of Buttazzo and Dal Maso and to write the restriction of F to W 1,1 (Ω; Rk ) as an integral; 7.4: the use of a relaxation result of Ambrosio and Dal Maso together with the formula (60) and some suitable estimates on the above integrand functions will finally give F ≤ H on BV (Ω; Rk ). Let us state in the sequel the integral representation theorem for functionals defined on Sobolev spaces we will use. Moreover let us recall a lower semicontinuity and relaxation theorem for quasiconvex integrals on the space of vector-valued BV -functions (this last result is a generalization to quasiconvex functions of the result by Goffman and Serrin used in Section 5 and Section 6). Theorem 22 (Integral representation theorem on Sobolev spaces). (see [28]) Let F : W 1,1 (Ω; Rk ) × A(Ω) → [ 0, +∞[ be a functional satisfying for every u , v ∈ W 1,1 (Ω; Rk ) and for every A ∈ A(Ω) the following properties: |∇u(x)| dx) ; (i) (linear growth) |F (u, A)| ≤ c(|A| + A

(ii) (locality) F (u, A) = F (v, A) whenever u = v on A ; (iii) (semicontinuity) F (·, A) is W 1,1 -sequentially lower semicontinuous; (iv) (translation invariance) F (u + b) = F (u, A) for every constant vector b ∈ Rk ; (v) F (u, ·) is the restriction to A(Ω) of a regular Borel measure. Then there exists a Carathéodory function ψ : Ω × Mk×n → [ 0, +∞[ , quasiconvex in the second variable for a.e. x ∈ Ω, such that the integral representation ψ(x, ∇u(x)) dx F (u, A) = A

holds for every u ∈ W

1,1

(Ω; R ) and for every A ∈ A(Ω). k

Theorem 23 (Semicontinuity on BV (Ω; Rk ) - Relaxation). (see [12]) Let ϕ : Mk×n → [ 0, +∞[ be a quasiconvex function satisfying 0 ≤ ϕ(ξ) ≤ c(1 + |ξ|)

∀ξ ∈ Mk×n .

Let F : BV (Ω; Rk ) → [ 0, +∞ ] be the functional defined by Ds u |Ds u| . F(u) = ϕ(∇u(x)) dx + ϕ∞ |Ds u| Ω Ω

(62)

(63)

Then (i) F is L1 (Ω; Rk )-lower semicontinuous on BV (Ω; Rk ) ; (ii) F = F + χW 1,1 (Ω;Rk ) , i.e., F coincides with the relaxation of its restriction to the Sobolev space W 1,1 (Ω; Rk ).


119

Remark 28. Note that in order to have a good definition of F in (59) and of F in (63) we have to extend the notion of ϕ∞ to ϕ quasiconvex. This quantity ϕ(tξ) if rank(ξ) ≤ 1, since quasiconvex functions are is well-defined by lim t→+∞ t convex in rank-one directions. In general, the above limit does not exist for all ξ ∈ Mk×n (see [42]). A more recent result by Alberti (see [2]) assures that Ds u is of rank 1 |Ds u|-a.e. , and hence F and F in (59) and in the matrix |Ds u| (63) respectively, are well defined. 7.1 Upper bound for the relaxed functional We start by giving an upper bound for the relaxed functional F . We shall consider the functional G : BV (Ω; Rk ) → [ 0, +∞ ] defined as ⎧ ⎪ ⎪ f (∇u(x)) dx + g((u+ − u− ) ⊗ νu ) dH n−1 ⎪ ⎪ ⎨ Ω Su ∩Ω if u ∈ SBV (Ω; Rk ) and H n−1 (Su ∩ Ω) < +∞, G(u) = ⎪ ⎪ ⎪ ⎪ ⎩ +∞ otherwise on BV (Ω; Rk ). (64) Clearly, F ≤ G. Obviously an upper bound of G = (relaxed functional of G with respect to the L1 -topology in BV ) will do as well. Proposition 4 (linear growth). We have G(u) ≤ c(1 + |Du|(Ω))

∀u ∈ BV (Ω; Rk ) .

we set Proof. Since g is locally bounded on Mk×n 1 M = sup{g(a ⊗ ν) : a ∈ S k−1 , ν ∈ S n−1 } < +∞ .

(65)

Step 1: k = 1. In this case g is defined on the whole Rn = R ⊗ Rn . Let us consider u ∈ BV (Ω) ∩ C 1 (Ω). Let us fix h ∈ N. By the co-area formula (10) we have j+1 h H n−1 (∂ ∗ {u > t} ∩ Ω) dt . |Du|(Ω) = j∈Z

j h

Hence, by the mean value theorem, for every j ∈ Z we can find shj ∈

& j j + 1' , h h

such that 1 n−1 ∗ H (∂ {u > shj } ∩ Ω) ≤ h

j+1 h j h

H n−1 (∂ ∗ {u > t} ∩ Ω) dt ,

120


so that

1 H n−1 (∂ ∗ {u > shj } ∩ Ω) ≤ |Du|(Ω) . h

(66)

j∈Z

Let us now construct the sequence (uh ) in SBV (Ω) as follows: uh (x) =

j h

on {shj−1 shj } ∩ Ω , Suh ∩ Ω = j∈Z

and Duh = Ds uh =

1 j ν H n−1 h h

∂ ∗ {u > shj } ∩ Ω ,

j∈Z

where νhj is defined by D1{u>shj } = νhj H n−1

∂ ∗ {u > shj } .

Hence we obtain − n−1 f (∇uh (x)) dx + g((u+ F (uh ) = h − uh ) ⊗ νuh ) dH Ω S ∩Ω u h 1 = f (0)|Ω| + g( νhj ) dH n−1 h h ∗ j∈Z ∂ {u>sj }∩Ω 1 M H n−1 (∂ ∗ {u > shj } ∩ Ω) ≤ f (0)|Ω| + h j∈Z

≤ f (0)|Ω| + M |Du|(Ω) . We have made use of the positive homogeneity of g, of (65) and (66). Remark also that by (66) for every h ∈ N we get H n−1 (∂ ∗ {u > shj } ∩ Ω) ≤ h|Du|(Ω) < +∞; H n−1 (Suh ∩ Ω) = j∈Z

hence

|Duh |(Ω) =

Suh ∩Ω

− n−1 |u+ = h − uh | dH

1 n−1 H (Suh ∩ Ω) ≤ |Du|(Ω) h

and G(uh ) = F (uh ) (note that we have proved H n−1 (Suh ∩ Ω) < +∞). Since (uh ) converges to u in L∞ (Ω), by the definition of G we conclude that G(u) = lim inf G(uh ) ≤ f (0)|Ω| + M |Du|(Ω) . h→+∞


121

For a general u ∈ BV (Ω) it suffices to recall that there exists a sequence (vh ) in C ∞ (Ω) ∩ BV (Ω) such that vh → u in L1 (Ω) and |Du|(Ω) = lim |Dvh |(Ω) = lim |∇vh | dx . h→+∞

h→+∞

Ω

By the lower semicontinuity of G we get G(u) ≤ lim inf G(vh ) ≤ lim f (0)|Ω|+M |Dvh |(Ω) = f (0)|Ω|+M |Du|(Ω) . h→+∞

h→+∞

Step 2: k ≥ 2. We can proceed “componentwise”: given a function u = (u(1) , . . . , u(k) ) ∈ C 1 (Ω; Rk ) ∩ BV (Ω; Rk ) we apply the same procedure as in Step 1 to every h ∈ N, j ∈ Z and i = 1, . . . , k (for more details see [22], Proposition 3.1). We obtain then by approximation √ G(u) ≤ f (0)|Ω| + kM |Du|(Ω) for every u ∈ BV (Ω; Rk ).

Remark 29. Let us point out that in Step 1 of the previous proof we have shown that fixed u ∈ BV (Ω) ∩ C 1 (Ω) there exists a sequence (uh ) ⊂ SBV (Ω) such that uh → u in L∞ (Ω), ∇uh = 0 a.e. on Ω, H n−1 (Suh ∩ Ω) < +∞, and |Duh |(Ω) ≤ |Du|(Ω). By using the density of C 1 (Ω) ∩ BV (Ω) in BV (Ω) and a diagonalization argument we can then assert that for every u ∈ BV (Ω) there exists a sequence (uh ) ⊂ SBV (Ω) such that uh → u in L1 (Ω), with ∇uh = 0 a.e. on Ω, H n−1 (Suh ∩ Ω) < +∞, and lim |Duh |(Ω) = |Du|(Ω). h→+∞

7.2 De Giorgi-Letta criterion to prove measure properties of the relaxed functional Our goal is to apply to the “localized version” of the functional G the integral representation theorem 22. The main difficulty is now to prove hypothesis (v) of Theorem 22 for G(u, ·). To this end we will use the following theorem by De Giorgi and Letta, which provides a sufficient condition to obtain a (regular) Borel measure starting with a (regular) increasing set function defined only on A(Ω). Theorem 24 (De Giorgi and Letta criterion). (see [35]) Let µ : A(Ω) → [ 0, +∞ ] be a positive set function defined on A(Ω). Assume (i) µ(∅) = 0, µ(A) ≤ µ(B) if A ⊆ B (µ is an increasing set function); (ii) µ(A ∪ B) ≤ µ(A) + µ(B) ∀A , B ∈ A(Ω) (µ is subadditive); (iii)µ(A ∩ B) ≥ µ(A) + µ(B) ∀A , B ∈ A(Ω) with A ∩ B = ∅ (µ is superadditive on disjoint sets); (iv) µ(A) = sup{µ(B) : B ⊂⊂ A , B ∈ A(Ω)} (µ is regular).

122


Then the extension µ to B(Ω) given by µ(B) = inf{µ(A) : A ∈ A(Ω) , B ⊆ A}

(68)

is a positive (regular) Borel measure. For a proof of the above theorem we refer also to [33], Theorem 14.23, [13], Theorem 1.53, [23], Theorem 10.2. Let us finally recall the Fleming-Rishel co-area formula. Let u be a Lipschitz function; then for every v ∈ BV (Ω) we have +∞ v|∇u| dx = v% dH n−1 dt (69) Ω

−∞

∂ ∗ {u>t}∩Ω

(∇u is the a.e. gradient of u, and v% denotes the approximate limit of v). Let us localize the functional G introduced by (64). We consider G(u, A) : BV (Ω; Rk ) × A(Ω) → [ 0, +∞ ] given by ⎧ ⎪ ⎪ f (∇u(x)) dx + g((u+ − u− ) ⊗ νu ) dH n−1 ⎪ ⎪ ⎨ A Su ∩A if u ∈ SBV (Ω; Rk ) and H n−1 (Su ∩ Ω) < +∞, G(u, A) = ⎪ ⎪ ⎪ ⎪ ⎩ +∞ otherwise on BV (Ω; Rk ). (70) Obviously, G(u, Ω) = G(u). We define also G(u, A) = inf{lim inf G(uh , A) : uh → u in L1 (A; Rk ),uh ∈ BV (Ω; Rk )} . h→+∞

Localizing the proof of Proposition 4 we get the linear growth condition G(u, A) ≤ c(|A| + |Du|(A))

(71)

for every u ∈ BV (Ω; Rk ), and for every A ∈ A(Ω). Proposition 5. For every u ∈ BV (Ω; Rk ) the set function G(u, ·) is (the restriction to A(Ω) of ) a regular Borel measure on Ω. Proof. Step 1. It is immediate to show that G(u, ·) is a positive, increasing set function. Step 2. G(u, ·) is regular; i.e., for every open set A ⊂ Ω, we have G(u, A) = sup{G(u, A ) : A open , A ⊂⊂ A} .

(72)

Indeed, since G(u, ·) is an increasing function, the inequality “ ≥ ” in (72) is trivial. Let us now prove the opposite inequality. Fixed a compact set K in A, let us define


δ= and

1 dist(∂A, K) 2

123

dK (x) = dist(x, K)

Bt = {x ∈ A : dK (x) < t} with t ∈ ]0, δ[ B = Bδ = {x ∈ A : dK (x) < δ} .

Note that for a.e. t ∈ ]0, δ[ , Bt is a set of finite perimeter. By definition of relaxation, there exist two sequences (uh ), (vh ) ∈ SBV (Ω; Rk ) such that H n−1 (Suh ∩ Ω) < +∞, H n−1 (Svh ∩ Ω) < +∞, and uh → u vh → u

in L1 (B; Rk ) in L1 (A \ K; Rk )

and G(u, A \ K) = lim G(vh , A \ K) .

G(u, B) = lim G(uh , B) h→+∞

h→+∞

Since we have H n−1 (Suh ∩ Ω) + H n−1 (Svh ∩ Ω) < +∞, we get that H n−1 (Suh ∩ ∂ ∗ Bt ) + H n−1 (Svh ∩ ∂ ∗ Bt ) = 0

(73)

for a.e. t ∈ ]0, δ[ (for a proof see Appendix B). We can apply the FlemingRishel co-area formula (69) with v = |uh − vh | and u(x) = dK (x) (recall that |∇dK (x)| = 1 a.e.); we get

|uh − vh | dx =

B\K

0

δ

∂∗B

|% uh (x) − v%h (x)| dH n−1 (x) dt .2 t

By the mean value theorem (see 3 below), for every h ∈ N we can choose th ∈ ]0, δ[ such that (73) holds, Bth is a set of finite perimeter, and 1 |% uh − v%h | dH n−1 ≤ |uh − vh | dx . (74) δ B\K ∂ ∗ Bt Let us now construct a “suitable” sequence wh in order to estimate G(u, A). We can define the sequence (wh ) in L1 (Ω; Rk ) by setting ⎧ ⎨ uh in Bth wh = ⎩ vh in Ω \ Bth (It is convenient to make a “cut” between minimizing sequences! Since Bth is a set of finite perimeter we have in particular H n−1 (∂ ∗ Bth ) = 0, hence 2

3

The functions u %h and % vh are the approximate limits of uh and vh , respectively. They exist H n−1 -a.e. on ∂ ∗ Bt since (73) holds. b 1 Let f ∈ L1 (a, b); then Ln ({t ∈ ]a, b[ : f (t) ≤ b−a f (s) ds}) > 0. a

124


Ln (∂ ∗ Bth ) = 0. Therefore wh is defined a.e. on Ω). Note that for every h we have H n−1 (Swh ∩ Ω) < +∞ . wh ∈ SBV (Ω; Rk ) , Moreover on A we have ∇wh = ∇uh 1Bth + ∇vh 1A\Bth , and the Hausdorff part of the measure Dwh is given by − n−1 (u+ h − uh ) ⊗ νuh H

Suh ∩ Bth +

+(vh+ − vh− ) ⊗ νvh H n−1

Svh ∩ (A \ Bth )+ ∂ ∗ Bth ,

+(% uh − v%h ) ⊗ ν∂ ∗ Bth H n−1

where ν∂ ∗ Bth denotes the normal to ∂ ∗ Bth pointing inwards Bth (by (73) we do not have jumps for wh on ∂ ∗ Bth ). We then obtain (using (65), the positive 1-homogeneity of g and (74)) B th ) G(wh , A) = G(wh , Bth ) + G(wh , A \ ≤ G(uh , B) + G(vh , A \ K) +

∂ ∗ Bth

≤ G(uh , B) + G(vh , A \ K) + M ≤ G(uh , B) + G(vh , A \ K) + M ≤ G(uh , B) + G(vh , A \ K) +

M δ

g((wh+ − wh− ) ⊗ ν∂ ∗ Bth ) dH n−1

∂∗B

|wh+ − wh− | dH n−1 th

∂ ∗ Bth

|% uh − v%h | dH n−1

|uh − vh | dH n−1 . B\K

Since wh → u in L1 (A; Rk ) and (uh − vh ) → 0 in L1 (B \ K; Rk ), we have, by taking the limit as h → +∞, G(u, A) ≤ lim inf G(wh , A) ≤ G(u, B) + G(u, A \ K) . h→+∞

By (71) we have then G(u, A) ≤ G(u, B) + c(|A \ K| + |Du|(A \ K)) . Since the last term in this inequality can be taken arbitrarily small, and B ⊂⊂ A, we get the desired equality in (72). Step 3. We shall prove that G(u, ·) is a subadditive set function on A(Ω). By the regularity of G(u, ·) (Step 2) it is sufficient to prove that G(u, A) ≤ G(u, A1 ) + G(u, A2 ) for every open set A ⊂⊂ A1 ∪ A2 , with A1 and A2 in A(Ω). This inequality can be proved by arguing as in Step 2, choosing K = A \ A2 and


B = {x ∈ Rn : dK (x)
0, we can consider the linear function ut (x) = tax, ν , so that we have Dut = ta ⊗ ν. We can approximate ut in L1 (Ω; Rk ) with a sequence (uht ) in SBV (Ω; Rk ) defined by uht (x) =

& 1 ' ta hx, ν h

which has jumps of size 1/h in the direction a, along hyperplanes orthogonal to ν at a regular distance of 1/h, and ∇uht = 0 a.e. ([s] is the integer part of s). Let Qν be any cube contained in Ω with an edge parallel to ν ; so we get ((uht )+ − (uht )− ) ⊗ νuht H n−1

Suht ∩ Qν =

1 ta ⊗ νH n−1 h

Suht ∩ Qν .

We have then (using the homogeneity of g) G(uht , Qν ) = f (0)|Qν | + g h1 ta ⊗ ν H n−1 (Suht ∩ Qν ) ≤ f (0)|Qν | + tg(a ⊗ ν)|Qν | .


127

Hence we obtain ψ(ta ⊗ ν) =

1 1 lim inf G(uht , Qν ) G(ut , Qν ) ≤ |Qν | |Qν | h→+∞

≤ f (0) + tg(a ⊗ ν) . Dividing by t, and letting t → +∞, we obtain ψ ∞ (a ⊗ ν) ≤ g(a ⊗ ν) , which is inequality (78). Proof of the relaxation theorem 21. Let H : BV (Ω; Rk ) → [ 0, +∞ ] be the functional on the right-hand side of (59), i.e., Ds u |Ds u| , ϕ(∇u(x)) dx + ϕ∞ (79) H(u) = |Ds u| Ω Ω where the function ϕ : Mk×n → [ 0, +∞[ is given by }. ϕ(ξ) = sup{ψ(ξ) : ψ quasiconvex, ψ ≤ f on Mk×n , ψ ∞ ≤ g on Mk×n 1 Step 1. H ≤ F on BV (Ω; Rk ). By the definition of ϕ we get immediately that 0 ≤ ϕ(ξ) ≤ f (ξ)

∀ξ ∈ Mk×n ;

moreover, it is easy to see that ϕ∞ ≤ g

on Mk×n . 1

Therefore H ≤ F on BV (Ω; Rk ). If we show that H is L1 -lower semicontinuous on BV (Ω; Rk ) we get H ≤ F on BV (Ω; Rk ). We will apply the semicontinuity theorem 23 by Ambrosio and Dal Maso; we have to prove that 0 ≤ ϕ(ξ) ≤ c(1 + |ξ|)

∀ξ ∈ Mk×n .

It suffices to show that φ(ξ) ≤ c(1 + |ξ|)

∀ξ ∈ Mk×n

for every quasiconvex function φ such that φ ≤ f on Mk×n and φ∞ ≤ g on . Let us fix such a φ. Since φ is quasiconvex, it is rank-one convex, which Mk×n 1 implies that for every ξ ∈ Mk×n , a ∈ Rk , b ∈ Rn the function t −→ φ(ξ + ta ⊗ b) is convex on R. In particular we get (using a truncation argument) that

128


φ(ξ + a ⊗ b) ≤ φ(ξ) + φ∞ (a ⊗ b) . n

Since every ξ ∈ Mk×n can be decomposed as ξ =

(80) am ⊗ em for suitable

m=1 n

am ∈ Rk ((em ) stands for the standard basis of R ), we get by applying repeatedly (80) φ(ξ) = φ(

n

m=1 n−1

≤ φ(

am ⊗ em ) = φ(

n−1

am ⊗ em + an ⊗ en )

m=1

am ⊗ em ) + φ∞ (an ⊗ en ) ≤ φ(0) +

m=1

≤ f (0) +

n

g(am ⊗ em ) ≤ f (0) + M

m=1

n

n

φ∞ (am ⊗ em )

m=1

|am ⊗ em | ≤ c(1 + |ξ|) ,

m=1

where M is defined by (65). Finally, ϕ is quasiconvex and satisfies (62) and we conclude that H is L1 -lower semicontinuous on BV (Ω; Rk ). Step 2. H ≥ F on BV (Ω; Rk ). By the definition of F , by applying the relaxation result in Theorem 23 we get for every u ∈ BV (Ω; Rk ) Ds u |Ds u| . F (u) ≤ G ≤ G + χW 1,1 (Ω;Rk ) = ψ(∇u(x)) dx + ψ∞ |Ds u| Ω Ω Applying Proposition 7 and the definition of ϕ in (60) we have that ψ ≤ ϕ and we get Ds u |Ds u| = H(u) F (u) ≤ ϕ(∇u(x)) dx + ϕ∞ |Ds u| Ω Ω for u ∈ BV (Ω; Rk ).

Additional remarks Remark 30. If f is positively 1-homogeneous, then ϕ is positively 1-homogeneous. In fact, it is immediate to check that ψ is a quasiconvex function such that ψ ≤ f on Mk×n and ψ ∞ ≤ g on Mk×n if and only if for every fixed λ > 0 1 1 such is the function φ(ξ) = ψ(λξ). We then obtain λ ϕ(ξ) = sup{ψ(ξ) : ψ quasiconvex, positively 1-homogeneous, ψ ≤ f on Mk×n , } ψ ∞ ≤ g on Mk×n 1 = sup{ψ(ξ) : ψ quasiconvex, positively 1-homogeneous, ψ ≤ f ∧ g on Mk×n } (since ψ = ψ ∞ for ψ positively 1-homogeneous; we have extended g = +∞ ). Anyhow, M¨ uller has shown in [42] that quasiconvexity and on Mk×n \ Mk×n 1 positive 1-homogeneity does not imply convexity.


129

Remark 31. Partitions. Let us consider the case 0 if ξ = 0, f (ξ) +∞ otherwise . Then the functional F < +∞ only on functions in the space SBV (Ω; Rk ) with ∇u = 0 a.e. These functions can be identified with “partitions of Ω in sets of finite perimeter” (see [9], [10], [29] ). Every such function can be expressed as cj 1Ej , u= j∈N

where cj ∈ Rk , and (Ej ) is a partition of Ω in sets of finite perimeter. The functional can then be written in the form 1 g((cj − ci ) ⊗ νj ) dH n−1 , (81) F (u) = 2 (∂ ∗ Ei ∩∂ ∗ Ej )∩Ω i,j∈N

where νj is the interior normal to Ej , and ∂ ∗ Ej denotes the reduced boundary of Ej . Since f is positively 1-homogeneous, by the previous remark, the relaxed functional F is given by Ds u |Ds u| F (u) = ϕ(∇u) dx + ϕ |Ds u| Ω Ω with } ϕ(ξ) = sup{ψ(ξ) : ψ quasiconvex, positively 1-homogeneous, ψ ≤ g on Mk×n 1 (ϕ = ϕ∞ since ϕ is positively 1-homogeneous). Note that if in particular g is quasiconvex then the functional (81) defined on partitions is lower semicontinuous with respect to the L1 -convergence.

8 Relaxation in higher dimension (isotropy assumptions) In this section we state a relaxation result for functionals defined on SBV (Ω) with a more general surface energy density as the functionals defined in (43) but with isotropy assumptions on the bulk energy term. Before we state the relaxation result by [17] let us introduce the notion of inf-convolution. Let f1 , f2 : R → [ 0, +∞ ] be proper (that is, fi (x) < +∞ for at least on x) convex functions, then the inf-convolution of f1 and f2 is defined as (82) (f1 ∇ f2 )(z) = inf{f1 (x) + f2 (z − x) : x ∈ R} and it turns out to be a convex function (see [43]). Moreover, one can prove that if f1 and f2 are non-negative convex functions on R, with f1 (0) = 0 and f2 positively homogeneous of degree 1, then

130


(f1 ∇ f2 )(z) = (f1 ∧ f2 )∗∗ (z) . Moreover, if f is convex, f (0) = 0, and θ is subadditive, positive and θ(0) = 0, then ⎧ f ∇ θ0 = (f ∇ θ0 )∗∗ f ∇ f∞ = f ⎪ ⎪ ⎪ ⎪ ⎨ (83) f ∞ ∇ θ = sub (f ∞ ∧ θ) ⎪ ⎪ ⎪ ⎪ ⎩ ∞ (f ∇ θ)0 = f ∞ ∇ θ0 = (f ∇ θ0 )∞ . Theorem 25 (Relaxation in BV (Ω) - Isotropy assumptions: interaction between bulk and interfacial energy densities). Let f : R → [ 0, +∞ ] be a convex function with f (0) = 0 ; let g : R → [ 0, +∞ ] be lower semicontinuous with g(0) = 0, locally bounded and such that the map t → g(|t|) is subadditive, i.e., g(|t1 + t2 |) ≤ g(|t1 |) + g(|t2 |) for every t1 , t2 ∈ R. Let Ω be a bounded open subset of Rn . Let F : BV (Ω) → [ 0, +∞ ] be defined by ⎧ ⎨ f (|∇u|) dx + g(|u+ − u− |) dH n−1 if u ∈ SBV (Ω), F (u) = Ω Su ∩Ω ⎩ +∞ otherwise on BV (Ω). Then the relaxed functional of F with respect to the BV − w∗ topology on BV (Ω) can be written as F (u) = f1 (|∇u|) dx + c1 |Ds u| + g1 (|u+ − u− |) dH n−1 , Ω

where

Ω\Su

f1 = f ∇ g 0 g1 = g ∇ f ∞

Ω∩Su

(= (f ∧ g 0 )∗∗ ) c1 = f1∞ (1)

(= g10 (1)).

Remark 32. Let us point out that the proof of this result follows an analogous scheme as seen in Section 7 for the proof of Theorem 21. The estimate of the lower bound of the relaxed functional uses a lower semicontinuity theorem proved previously in the paper quoted above. The upper bound is obtained by using a measure theoretical approach as in Theorem 21, the relaxation result in Theorem 17 and a representation theorem on partitions (see [9]). Remark 33. The authors suppose 0 < min{f ∞ (1), g 0 (1)} < +∞. Let us note that in the case min{f ∞ (1), g 0 (1)} = +∞, we get f1 = f , g1 = g, c1 = +∞, and then F = F . This lower semicontinuity theorem is a result proven by [5], see Theorem 12 of these lecture notes. On the other hand, if 0 = min{f ∞ (1), g 0 (1)} then F = 0. Applications By applying the above theorem it follows immediately by easy calculations that with g(t) = |t|q , f (t) = |t|p we get that for


(a) 1 < p < +∞ and 0 < q < 1 |∇u|p dx + F (u) = Ω

Su ∩Ω

131

|u+ − u− |q dH n−1

(b) p = 1 and q = 1

|∇u| dx +

F (u) = Ω

Su ∩Ω

|u+ − u− | dH n−1

are BV − w∗ lower semicontinuous functionals on SBV (Ω). Furthermore, if (c) p > 1 and q = 1 f1 (ξ) = (|ξ|p ∧ |ξ|)∗∗ = g1 (ξ) = |ξ|

|ξ|p p |ξ| − (p − 1)p 1−p

1

if |ξ| ≤ p 1−p 1 if |ξ| > p 1−p

c1 = 1 ,

(d) p = 1 and 0 < q < 1 f1 (ξ) = |ξ| |ξ| g1 (ξ) = |ξ|q

if |ξ| ≤ 1 if |ξ| ≥ 1

c1 = 1 .

For further applications see [26]. Remark 34. (see [26]) Some interesting examples do not fall into the known framework. Consider for instance a bulk energy density f (z) with growth p > 1 (as in the case of linear elasticity, where p = 2) and a fracture energy density g of the form ⎧ ⎪ if s, ν > 0 ⎨1 g(s, ν) = +∞ if s, ν < 0 ⎪ ⎩ γ(s) if s, ν = 0, where γ is a function with linear growth. The term 1 describes the energy necessary to produce a normal detachment according to the Griffith model, the term +∞ is to avoid the interpenetration of the two sides of the fracture, and the term γ(s) takes into account the tangential sliding. It would be interesting to compute the relaxed functional associated to the model above, and to see if the formulas for f1 and g1 obtained in Theorem 25 still hold. Let us mention that in [24] we consider the case when also normal detachments are forbidden (i.e., g = +∞ if s, ν = 0), taking ⎧ ⎨ β|s| √ if s, ν = 0 1 D 2 2 g(s, ν) = f (z) = |e (z)| + α|tr(z)| , 2 ⎩ 2 +∞ otherwise,

132


where eD (z) is the deviator of the symmetric part of z defined by eD (z) = tr(z) z + zt − I. We obtained the relaxed integrands 2 n g(s, ν) = ϕ∞ (eD (s ⊗ ν)) ,

f (z) = ϕ(eD (z)) + α|tr(z)|2

where ϕ is the conjugate function on the set of all symmetric matrices with null trace of the function √ 1 2 σ if λM (σ) − λm (σ) ≤ β, 2 h(σ) = +∞ otherwise being λM (σ), λm (σ) the maximum and minimum eigenvalue, respectively, of σ. In this setting let us also quote the paper by [25] on relaxation of elastic energies with free discontinuities and constraint on the strain. Finally I will mention the paper by [18], where the authors introduce a new method for the identification of the integral representation of functionals defined on BV (Ω; Rk ) × A(Ω). Some relaxation results in SBV (Ω; Rk ) are recovered in the case where bulk and surface energies are present.

9 Appendix 9.1 Appendix A: Lower semicontinuous envelope Subadditive envelope Definition. The subadditive envelope sub ϕ of ϕ : R → R is the greatest subadditive function less than or equal ϕ. It is easy to see that sub ϕ(x) = inf

m ! k=1

ϕ(xk ) :

m

" xk = x , m = 1, 2, . . . .

k=1

Definition. sub ϕ(x) = sub (sc− ϕ)(x) for every x ∈ R. Note. If φ : R × R → R and ϕ : R → R such that φ(x, y) = ϕ(x − y) then sub φ(x, y) = sub ϕ(x − y). A1. If ϕ : R → R is convex (hence continuous), then sub ϕ = sub ϕ can be computed more easily: " ! x : k = 1, 2, . . . . sub ϕ(x) = inf kϕ k k This follows immediately by the convexity inequality kϕ xk ≤ j=1 ϕ(yj ), k whenever y = j=1 yj .


133

A2. If ϕ : R → R is subadditive and locally bounded, then it grows less than linearly at infinity. Indeed, for every y ∈ R we have by subadditivity ϕ(y) = ϕ (1 + [|y|])

y y ≤ (1 + [|y|])ϕ , (1 + [|y|]) (1 + [|y|])

where [s] is the integer part of s. Since M = sup |ϕ(x)| < +∞ we get |x|≤1

ϕ(y) ≤ M (1 + |y|)

∀y ∈ R .

A3. Let ϕ : R → R be defined by ϕ(x) = 1 + x2 . By A2 we get that ϕ is not subadditive. Since ϕ is convex, by A1 we have " ! t2 : k = 1, 2, . . . . sub ϕ(x) = min k + k A4. Let ϕ : R → R be defined by ϕ(x) = (2|x| − 1) ∨ 1. Since ϕ is convex and even, by A1 we get that sub ϕ is even and continuous and in [ 0, +∞[ we have ⎧ ⎪ if x ≤ 1 ⎨1 sub ϕ(x) = k + 2(x − k) if k ≤ x ≤ k + 12 k = 1, 2, . . . ⎪ ⎩ k = 1, 2, . . . k+1 if k + 12 ≤ x ≤ k + 1 In this case we have |x| ≤ sub ϕ(x) ≤ |x| +

1 2

for |x| ≥ 12 . We have sub ϕ(x) = |x| for x = ±1 , ±2 , . . . , sub ϕ(x) = |x|+ 12 for x = ± 12 , ± 32 , . . . , and hence sub ϕ is not asymptotic to a linear function as x → ±∞. A5. Let ϕ : R → R be defined by ϕ(x) = |x − 1|. By A1 we have sub ϕ(x) = min{|x − k| : k = 1, 2, . . .} i.e.,

1−x sub ϕ(x) = dist(x, N)

if x ≤ 1 if x ≥ 1.

In this case we have that the limit lim sub ϕ(x) does not exist. x→+∞

Proof of Proposition 3. (i) Let λ = inf φ > 0. Then λ ≤ φ(x, y) for every x, y ∈ R. By the definition of sc− φ (= greatest lower semicontinuous function less than or equal φ, see Definition 4) we get λ ≤ (sc− φ)(x, y) ≤ φ(x, y) for all x, y ∈ R. Therefore, λ = inf(sc− φ) ≥ λ > 0. Being ψ(x, y) ≡ λ a subadditive function less than or equal φ(x, y) for every x, y ∈ R, by definition of subadditive envelope we get

134


0 < λ ≤ sub (sc− φ)(x, y) = sub φ(x, y) ≤ φ(x, y) for every x, y ∈ R. Hence inf sub φ > 0. From φ(x, y) ≥ |x − y| for every x, y ∈ R it follows that |x − y| ≤ (sc− φ)(x, y) ≤ φ(x, y) for all x, y ∈ R. Since ψ(x, y) = |x − y| is subadditive, by definition of subadditive envelope we get |x − y| ≤ sub φ(x, y) ≤ φ(x, y)

∀x, y ∈ R .

(ii) Let us prove that sub φ is lower semicontinuous. Fixed x, y ∈ R and two sequences xh → x, yh → y such that there exists the limit lim sub φ(xh , yh ), h→+∞

we have to prove that sub φ(x, y) ≤ lim sub φ(xh , yh ) . h→+∞

By definition for every h ∈ N there exist xh0 , . . . , xhmh such that xh0 = yh , xhmh = xh , and mh

(sc− φ)(xhk , xhk−1 ) ≤ sub φ(xh , yh ) +

k=1

1 h.

By the condition inf(sc− φ) > 0 we have that the sequence (mh ) is bounded. Hence we can suppose mh = m, independent of h. The condition (sc− φ)(x, y) ≥ |x − y| implies that all sequences (xh0 ) , . . . , (xhmh ) are bounded. Again we can suppose then that xh0 → x0 , xh1 → x1 , . . . , xhm → xm for some x0 , x1 , . . . , xm ∈ R. Of course, x0 = y, and xm = x. By semicontinuity of sc− φ we get (sc− φ)(xk , xk−1 ) ≤ lim inf (sc− φ)(xhk , xhk−1 ) , h→+∞

and hence we get m

(sc− φ)(xk , xk−1 ) ≤

k=1

m k=1

lim inf (sc− φ)(xhk , xhk−1 ) h→+∞

≤ lim inf

h→+∞

m

(sc− φ)(xhk , xhk−1 ) ≤ lim sub φ(xh , yh ) . h→+∞

k=1

By definition this shows that sub φ(x, y) ≤ lim sub φ(xh , yh ) . h→+∞

Suppose now that there exists φ1 : R × R → R lower semicontinuous and subadditive such that φ1 ≤ φ. We get then immediately that φ1 ≤ sc− φ ≤ φ (by definition of sc− φ). Furthermore, being φ1 subadditive φ1 ≤ sub (sc− φ) ≤ φ . This proves (ii).


135

9.2 Appendix B: Proof of (73) Let us show that for every h ∈ N, the set t ∈ ]0, δ[ that does not satisfy H n−1 (Suh ∩ ∂ ∗ Bt ) = 0 is at most countable. Let Th = {t ∈ ]0, δ[ : H n−1 (Suh ∩ ∂ ∗ Bt ) > h1 }. We claim that Th is finite. Indeed, if Th is countable we get n−1 H n−1 (Suh ∩ Ω) ≥ H n−1 Suh ∩ ∂ ∗ Bt = H (Suh ∩ ∂ ∗ Bt ) t∈Th

t∈Th

1 = +∞ . ≥ h t∈Th

Therefore Th has to be finite (since H n−1 (Suh ∩Ω) < +∞ holds). We conclude that Th T = {t ∈ ]0, δ[ : H n−1 (Suh ∩ ∂ ∗ Bt ) > 0} = h

is at most countable.

9.3 Appendix C: Proof of Step 3 in Proposition 67 Let us show that for every A1 , A2 ∈ A(Ω) G(u, A) ≤ G(u, A1 ) + G(u, A2 ) for every open set A ⊂⊂ A1 ∪ A2 . It is not restrictive to suppose G(u, Ai ) < +∞ , i = 1, 2. Fixed A ⊂⊂ A1 ∪ A2 , let us choose the compact set K = A \ A2 , and set δ = 12 dist(K, A \ A1 ) Bt = {x ∈ Rn : dist(x, K) < t}

for t ∈ ]0, δ[

B = Bδ .

Note that for a.e. t ∈ ]0, δ[ , Bt is a set of finite perimeter. Let us consider the minimizing sequences (uih ) ⊂ SBV (Ω; Rk , uih → u in L1 (Ai ; Rk ) with H n−1 (Suih ∩ Ω) < +∞ and G(u, Ai ) = lim G(uih , Ai ) h→+∞

i = 1, 2 .

Since we have H n−1 (Suih ∩ Ω) < +∞, we get H n−1 (Suih ∩ Ω ∩ ∂ ∗ Bt ) = 0

(84)

for a.e. t ∈ ]0, δ[ (see Appendix B). By the Fleming-Rishel co-area formula and the mean value theorem, for every h ∈ N, we can choose th ∈ ]0, δ[ such that (84) holds, Bth is a set of finite perimeter, and

136


∂ ∗ Bth ∩A

|% u1h

−

u %2h | dH n−1

1 ≤ δ

|u1h − u2h | dx ,

(85)

A

where u %ih are the approximate limits of uih (they exist H n−1 -a.e. on ∂ ∗ Bth by (84)). Let us define the sequence u1h on Ω ∩ Bth wh = u2h on Ω \ Bth . Then wh ∈ SBV (Ω; Rk ), H n−1 (Swh ∩ Ω) < +∞, and wh → u in L1 (A; Rk ). Moreover, on A we have ∇wh = ∇u1h 1A∩Bth + ∇u2h 1A\Bth and the Hausdorff part of the measure Dwh is given by ((u1h )+ − (u1h )− ) ⊗ νu1h H n−1

Su1h ∩ A ∩ Bth +

+ ((u2h )+ − (u2h )− ) ⊗ νu2h H n−1 %2h ) ⊗ ν∂ ∗ Bth H n−1 + (% u1h − u

Su2h ∩ (A \ Bth )+

∂ ∗ Bth ∩ A ,

where ν∂ ∗ Bth is normal to ∂ ∗ Bth pointing inwards Bth . Then (using the positive 1-homogeneity of G, (65) and (85)) G(wh , A) ≤ G(u1h , A ∩ Bth) + G(u2h , A \ Bth )+ 1 g (% uh − u %2h ) ⊗ ν∂ ∗ Bth dH n−1 + ∂ ∗ Bth ∩A 1 ≤ G(uh , A1 ) + G(u2h , A2 ) + M |u1h − u2h | dx . δ A

By taking the limit as h → +∞, we get G(u, A) ≤ lim inf G(wh , A) ≤ G(u, A1 ) + G(u, A2 ) . h→+∞

as desired.

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Convergence of Dirichlet forms on fractals Roberto Peirone Dipartimento di Matematica, Universit` a di Roma “Tor Vergata” via della Ricerca Scientifica, 00133, Roma, Italy e-mail: [email protected]

1 Introduction The purpose of these notes is to introduce the theory of the convergence, and specially, to discuss the Γ -convergence, of Dirichlet forms on fractals. A Dirichlet form is a sort of “energy”, in some sense it is a generalization of the Dirichlet integral u → |grad u|2 defined for u in open regions in Rn . The investigation of Dirichlet forms on fractals began in connection with the construction of a Brownian motion on fractals. It is also closely related to the construction of a Laplacian or in other words to the definition of harmonic functions on a fractal. In some sense, on fractals, the notions of Dirichlet forms, Brownian motion, and harmonic functions are, in fact, three different points of view of the same notion. In these Notes I will discuss the notion of a Dirichlet form, as well as that of a harmonic function, as we will need it in connection with some properties of Dirichlet forms. On the contrary, I will not discuss the notion of a Brownian motion. The class of fractals which I discuss here is that of finitely ramified fractals. More or less we are in the following situation. We are given finitely many similarities in Rν with ν ≥ 1, which we denote by ψ1 , ..., ψk . Suppose they are contractions, i.e., their factors r1 , ..., rk are < 1. Then, there exists a unique nonempty compact K in Rν such that k # K= ψi (K) (see [4]). We will say that such a set K is the fractal generated i=1

by the set of similarities. The finite ramification means, more or less, that different copies ψi (K) of K can have at most finitely many (in general at most one) common points. This class of fractals includes for example the Gasket, but not the Carpet. The construction of a Dirichlet form on fractals that are not finitely ramified is much more complicated and will be not discussed in these Notes. A finitely ramified fractal can also be seen as the closure of the union of an increasing sequence of finite sets V (0) , ..., V (n) , .... In defining a Dirichlet form on the fractal, we meet the problem that usually, K has an empty interior, hence we cannot define the gradient on it, at least in the usual sense. So, we construct a Dirichlet form on K, by taking the limit of discrete

140

Roberto Peirone

Dirichlet forms on V (n) . A nontrivial point in this construction is the existence of such a limit form. However, if the initial form E on V (0) is an eigenvector of a particular nonlinear minimization operator, also called renormalization, in our terms if E is an eigenform, then we can define a sequence of discrete Σ , that are in some sense the sum of the copies of E on the Dirichlet forms E(n) n-cells, suitably renormalized. Now, for every function v defined on K, the Σ (v) is increasing and thus has a limit which we will denote by sequence E(n) Σ E(∞) (v). Such a limit form turns out to be a Dirichlet form on K. Sections 2 to 4 are introductory and recall the construction and the main properties of a Dirichlet form on finitely ramified fractals. For the general theory of Dirichlet forms the reader can refer to [2]. The construction of Dirichlet forms on finitely ramified fractals is described for example in [1], [8], which specially stress the probabilistic point of view, and in [6] which instead deals with Dirichlet forms and Laplacian, in a very general class of fractals, called p.c.f. self-similar sets, introduced by J. Kigami in [5]. The approach followed here is, in some sense, similar to that in [6], but, in order to simplify the presentation, I have preferred not to give the general definition of Dirichlet form, but only to discuss a particular kind of Dirichlet form. An even more similar approach to that presented here can be found in [20]. I have preferred to start, in Section 2, with a specific example, the Sierpinski Gasket, as, in my opinion it allows us to better understand the problem of the construction of a Dirichlet form in a simple case. In Section 3, I first give a general definition of finitely ramified fractals, and then I discuss in such a class the construction of a Dirichlet form. While in the case of Gasket, by symmetry, we explicitly found an eigenform, in the general case of finitely ramified fractals, the existence of an eigenform is a nontrivial problem, and there are finitely ramified fractals having no eigenforms. However, the class of finitely ramified fractals having at least an eigenform includes several important examples of finitely ramified fractals, in particular, the nested fractals introduced by T. Lindstrøm [9], and we will only restrict our considerations to such a class of fractals. Section 4 is devoted to the investigation of the properties of the renormalization operator and of the related notion of the harmonic extension. The actual aim of these Notes, i.e., the convergence of Dirichlet forms on finitely ramified fractals, is described in Sections 5 and 6. While Sections 1 to 4 mainly describe known results by other authors, Sections 5 and 6 mainly treat results of the author, specially in [16]. When E is not an eigenform we cannot use the same process as for the eigenforms to construct a Dirichlet form starting with E, since the corresponding sequence of discrete forms is no longer increasing. It could be proved that nevertheless, the discrete forms pointwise converge to a Dirichlet form defined on K, but the proof is more complicated (see [17]), and will be not discussed here. I will discuss, instead, the Γ -convergence of the sequence of Σ Γ -converges to the Dirichlet form discrete forms. I will show that in fact E(n) ˜ ˜ as the limit of a sequence on K associated to an eigenform E. We will obtain E (0) n ˜ ˜ 1 is the renormalization of forms E(n) on V , defined as M1 (E) where M

Convergence of Dirichlet forms on fractals

141

˜ n (E) is the most operator divided by the eigenvalue ρ. The convergence of M 1 delicate point in the proof of Γ -convergence. It is, in some sense, a problem of convergence of the iterated of a map that it is known to have a fixed point. In the present case, the fixed point is the eigenform. In Section 5, I discuss the Γ -convergence on the Gasket. Due to the symmetry of the Gasket, the ˜ 1n (E) is much simpler than in the general case. proof of the convergence of M There are many different proofs of this result in the case of the Gasket. The proof presented here is probably not the simplest, but suggests the way of proceeding in the general case. Then, following an argument due to S. Kozlov [7], we deduce the Γ -convergence result. In Section 6, following [16], I prove ˜ 1n (E) in the general case. I restrict, in fact, the class of the convergence of M fractals in order to simplify the argument, avoiding some technical difficulties. At the end of the section, I merely hint the idea in the most general case. The class of fractals considered in the actual proof in Section 6, however, includes most of the usual fractals, thus it is sufficiently general for many purposes. The idea of the proof consists in proving that, with respect to Hilbert’s projective ˜1 ˜ 1n (E) get closer and closer. Note that in general M metric, the iterated M is not a contraction and in fact can have different fixed points. Once the ˜ n (E) is proved, the Γ -convergence result follows as in the convergence of M 1 case of the Gasket. As these Notes are not intended for specialists in fractals, I have tried to highlight the general ideas rather than the details. So, in some cases, for example for the notion of finitely ramified fractals, I have not used the most general definitions. For the same reason, I have preferred to give proofs which are not necessarily the shortest, but which require no non usually known results, like as the general theory of Hilbert’s projective metric, in order to make these Notes as self-contained as possible. I now fix some notation for the following. When we are in Rn , we will denote by d the euclidean distance, and unless specified otherwise, we will denote by || || the euclidean norm. Sometimes, we will use RA where A is a set. In such a case, of course, on RA , we put the norm and the metric, obtained by identifying RA with RM where M = #A. If f is a map from a set X into itself, we will denote by f n the nth -iterated of f , i.e., the composition of n maps equal to f . If X is a topological space, we will denote by C(X) the set of the continuous functions from X into R. If A is a subset of a euclidean space, we will denote by coA the convex hull of A.

2 Construction of an energy on the Gasket The first step in our work consists in constructing an “energy” on the fractals. Energy here, means an analog of the Dirichlet integral, u → |grad u|2 , on fractals. Probably, the reader has at least a rough idea of the notion of fractal. It is in some sense, a set that contains copies of itself, on arbitrarily small scales. In this section, in order to make these notions clear, I prefer to describe a specific example of fractal, the (Sierpinski) Gasket. In the next

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section I will give a precise notion of a fractal (or self-similar set). The Gasket, roughly speaking, can be constructed like the Cantor set, but starting from a triangle, instead of from a segment-line. More precisely, start with an equilateral triangle T , whose vertices are denoted by P1 , P2 , P3 , and consider the three rotation-free similarities ψi , i = 1, 2, 3, in R2 , that are contractions with factor 12 and have Pi as fixed points, in formula ψi (x) = Pi + 12 (x − Pi ). Then the (Sierpinski) Gasket is the set K defined by K0 = T, Kn+1 =

3

ψi (Kn ),

K=

∞ (

Kn ,

(1)

n=0

i=1

in other words, we split the initial triangle T into four similar triangles and remove the central one. Then, we remove the central triangle in each of the three remaining triangles. Then, we repeat the same process on each of the nine remaining triangles, and so on. In Figure 1, we depict K0 , K1 and K2 . P2

P12

P1

P23

P13

ø

P3

Fig. 1. The Sierpinski Gasket

In the√ previous construction, we can fix the vertices, e.g., P1 = (0, 0), P2 = ( 12 , 23 ), P3 = (1, 0), in order to define a precise triangle. Put now V = V (0) = {P1 , P2 , P3 },

Vi1 ,...,in = ψi1 ,...,in (V (0) )

for i1 , ..., in = 1, 2, 3, where ψi1 ,...,in is an abbreviation for ψi1 ◦ ... ◦ ψin , and put V (n) =

3 i1 ,..,in =1

Vi1 ,...,in ,

V (∞) =

∞

V (n) .

n=0

The sets Vi1 ,...,in are called n-cells and are, in some sense, small copies of V (0) , more precisely, the copies of V (0) at the nth step. They can, of course, also be interpreted as the sets of the vertices of the triangles ψi1 ,...,in (T ), which are copies of T at the nth step. More generally, put Ai1 ,...,in = ψi1 ,...,in (A) for every A ⊆ R2 . I will call Ai1 ,...,in an (nth A) copy. Clearly, we have


143

V ⊆K⊆T. Now, we want to define an “energy” on K. In this section, I will give no precise meaning to the word energy, and I will define it in the next section, when I treat the case of general fractals. Roughly speaking, we have to define an object that in some sense resembles the Dirichlet integral. However as the Gasket has an empty interior, it is not possible to define the gradient on it. The way of constructing an energy is based on a finite-difference scheme that I will now illustrate. Since it is easy to see that V (n) ⊆ V (n+1) , and, as we will see in the following, K is the closure of V (∞) , the idea consists in defining an energy first on V (0) , next on V (n) as the sum of the copies of it on all n-cells, and finally on all of K, taking a sort of limit. More precisely, let (u(Pj1 ) − u(Pj2 ))2 E(u) = 1≤j1 <j2 ≤3

for all u : V (0) → R. Note that E(u) ≥ 0, and the equality holds if and only if u is constant on V (0) . This is a trivial but crucial remark. Let S0 (E) = E 3

Sn (E)(v) =

E(v ◦ ψi1 ,...,in ) for v ∈ RV

(n)

,n ≥ 1.

i1 ,...,in =1

We can also write Sn (E)(v) =

2

v(Q) − v(Q )

for v ∈ RV

(n)

, n ≥ 1,

where the sum is extended over all pairs (Q, Q ) ∈ V (n) × V (n) which are close in the sense that they lie in a common n-cell. Now, we would like to define an energy E for functions defined on K as the limit of Sn (E), but some difficulties arise. First, this limit may not exist; moreover, it is 0 for too many functions, for example it is 0 on all the linear functions; we on the contrary require that it is 0 only for the constant functions. We are so lead to introduce a renormalization factor ρ, i.e., we take the limit of ρ1n Sn (E). This is a natural device in the sense that in classical cases, such as e.g., the Dirichlet integral we have to introduce a renormalization factor. In the present case, however, the value of ρ is not much expected. As we will see later, we have, in fact, ρ = 35 . In order to find the value of ρ, we now introduce a minimization operator. (0) Namely, for u : RV → R, let Mn (E)(u) = inf{Sn (E)(v) : v ∈ L(n, u)} ∀ u ∈ RV (n)

(0)

(2)

where L(n, u) = {v ∈ RV : v = u on V (0) }. We will see in the following that the infimum in (2) is in fact a minimum. In terms of potential theory, Mn (E) is the trace of Sn (E) on V (0) . For the moment let us study only M1 .

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Put P12 = ψ1 (P2 ) = ψ2 (P1 ), P13 = ψ1 (P3 ) = ψ3 (P1 ), P23 = ψ2 (P3 ) = ψ3 (P2 ). We thus have S1 (E)(v) = (v(P1 ) − v(P12 ))2 + (v(P1 ) − v(P13 ))2 + (v(P12 ) − v(P13 ))2 +(v(P2 ) − v(P12 ))2 + (v(P2 ) − v(P23 ))2 + (v(P12 ) − v(P23 ))2 +(v(P3 ) − v(P13 ))2 + (v(P3 ) − v(P23 ))2 + (v(P13 ) − v(P23 ))2 and M1 (E)(u) is the infimum of (u(P1 ) − x)2 + (u(P1 ) − y)2 + (x − y)2 +(u(P2 ) − x)2 + (u(P2 ) − z)2 + (x − z)2 +(u(P3 ) − y)2 + (u(P3 ) − z)2 + (y − z)2 for x, y, z ∈ R. As the function to minimize is convex, it attains its minimum at the points at which its gradient is 0. Hence, (x, y, z) is a minimum point if and only if ⎧ ⎪ ⎨ 4x = y + z + u(P1 ) + u(P2 ) (3) 4y = x + z + u(P1 ) + u(P3 ) ⎪ ⎩ 4z = x + y + u(P2 ) + u(P3 ) A simple calculation yields ⎧ 2u(P1 ) + 2u(P2 ) + u(P3 ) ⎪ ⎪ x= ⎪ ⎪ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ 2u(P1 ) + 2u(P3 ) + u(P2 ) y= ⎪ 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ z = 2u(P2 ) + 2u(P3 ) + u(P1 ) . 5 We will denote by H(1;E) (u) : V (1) → R the so obtained solution of (3), i.e., the unique function v ∈ L(1, u) such that M1 (E)(u) = S1 (E)(v). By substituting the value of v = H(1;E) (u), we get 3 E(u), 5 thus M1 (E) is a multiple of E. Is this a lucky case or not? To answer this question, observe that H(1;E) is linear, so M1 (E)(u) is quadratic in u, in other words it is a linear combination of terms of the form u(Pj )2 and of terms of the form u(Pj1 )u(Pj2 ). Moreover, since, clearly H(1;E) (u + c) = H(1;E) (u) + c, we can replace u(Pj ) by u(Pj ) − u(P1 ), and thus we have that M1 (E)(u) =


145

M1 (E)(u) = a(u(P2 ) − u(P1 )) + b(u(P3 ) − u(P1 ))2 + d(u(P2 ) − u(P1 ))(u(P3 ) − u(P1 )); 2

now, by the well-known formula αβ = 12 (α2 + β 2 − (α − β)2 ) with α = u(P2 ) − u(P1 ), β = u(P3 ) − u(P1 ), we have M1 (E)(u) = a (u(P2 ) − u(P1 ))2 + b (u(P3 ) − u(P1 ))2 + d (u(P2 ) − u(P3 ))2 for some a , b , d and by the symmetry of the Gasket we must have a = b = d (I here do not prove completely the last assertion, but a formal proof of it due to symmetry can be easily given). In conclusion, there is a natural reason for which M1 (E) is a multiple of E, and the previous calculation can be used to the only aim of evaluating the factor 35 . Note that, by the definition of M1 (E), we have S1 (E)(v) ≥ M1 (E)(v) for every v : V (1) → R where we identify a function v : V (1) → R with its restriction on V (0) , and we will do similarly in other cases. Hence, putting ρ = 35 we have M1 (E)(v) S0 (E)(v) S1 (E)(v) ≥ = E(v) = . ρ ρ ρ0

(4)

Suppose now v : V (∞) → R. We are going to prove that, more generally, the sequence Sn (E)(v) is increasing in n, thus it has a limit, which will be the ρn energy on K. To see this, note that 1 Sn+1 (E)(v) = n ρn+1 ρ

=

≥

1 ρn

1 ρn

i1 ,...,in =1

3 i1 ,...,in =1

3 i1 ,...,in =1

3 1

3

1

1 ρ

ρi

E(v ◦ ψi1 ,...,in ,in+1 )

n+1 =1

S1 (E)(v ◦ ψi1 ,...,in )

E(v ◦ ψi1 ,...,in ) =

1 Sn (E)(v) . ρn

In the above inequality we have used (4). Now put Σ E(n) (v) =

(n) 1 Σ Σ Sn (E)(v) for v ∈ RV , E(∞) (v) = lim E(n) (v) for v ∈ RK . n n→∞ ρ

Σ (v) can We have thus constructed an energy on K. Note that of course E(∞) amount to ∞. An unexpected fact is that the linear nonconstant functions

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Roberto Peirone

have infinite energy. In fact, let v : K → R be linear (or more generally affine). We easily get that E(v ◦ ψi1 ,...,in ) = ( 14 )n E(v), hence Sn (E)(v) = Σ Σ (v) = ( 54 )n E(v). It follows that E(∞) (v) = +∞ unless ( 34 )n E(v), and E(n) E(v) = 0, i.e., v is constant. Thus the set of the functions with finite energy is in some sense completely different from that in the case of a region in Rn . So, Σ is identically +∞. In the rest of this section one could even suspect that E(∞) we prove that this is not so. More precisely, we will prove the following two Σ . properties of E(∞) Σ a) E(∞) (v) = 0 if and only if v is constant on K. Σ b) The set of v ∈ C(K) such that E(∞) (v) < +∞ is dense in C(K).

In order to prove a), we need a Lemma. Σ Lemma 1. If v : V (n) → R and E(n) (v) = 0, then v is constant on V (n) .

Proof. By the definition of V (n) , we have V (m+1) =

3 #

ψi (V (m) ) for all m ∈ N.

i=1 (0)

Moreover, ψi (V (m) ) ∩ ψj (V (m) ) ⊇ ψi (V (0) ) ∩ ψj (V ) = ø. Thus, if n > 0 and v is nonconstant on V (n) , it is also nonconstant on ψi (V (n−1) ), or in other words, v ◦ ψi is nonconstant on V (n−1) , for some i = 1, 2, 3, and by a recursive argument, v ◦ ψi1 ,i2 ,...,in is nonconstant on V (0) for some i1 , i2 , ..., in = 1, 2, 3. Σ Σ , we have E(n) (v) > 0. Hence by the definition of E(n) Σ (v) = 0, then v is constant on K. Theorem 1. If v ∈ C(K) and E(∞) Σ (v) = 0 for each n ∈ N, hence v Proof. We deduce from the hypothesis, E(n)

is constant on V (∞) by Lemma 1. As K = V (∞) , v is constant on K.

We are now going to prove b). To do this, it is useful to consider the notion of the harmonic extension on V (1) . Given u : V (0) → R we know that the function v := H(1;E) (u) can be characterized as the unique solution of the system (3) where v(P12 ) = x, v(P13 ) = y, v(P23 ) = z. Such a function v is called the harmonic extension of u on V (1) . Note that (3) says that the value of v at a point Q of V (1) \ V (0) is the mean value of the values at the points of V (1) close to Q; this is an analogous property to the mean property of harmonic functions in regions in Rn . Another analogy is that harmonic functions also are a sort of minimum of the Dirichlet integral. Note that we have Σ (H(1;E) (u)) = E(u) (5) E(1) We will use the harmonic extension to construct a function v defined on all K, by extending harmonically u on V (1) , next applying this process on every 1-cell to obtain a function defined on V (2) , next applying the same process on every 2-cell and so on. However, in order to know that in this way we have in fact defined a continuous function, we have to prove that the oscillation on


147

n-cells tends to 0. The harmonic extension satisfies the following maximum principle: Lemma 2. For every u ∈ RV

(0)

we have

min u ≤ H(1;E) (u)(Q) ≤ max u V (0)

V (0)

∀ Q ∈ V (1) \ V (0)

(weak maximum principle). Moreover, the inequalities are strict unless u is constant (strong maximum principle). Proof. It suffices to prove the first inequality, the proof of the second being analogous. Put v = H(1;E) (u). If v does not attain its minimum at any point of V (1) \ V (0) , then we have min v = v(Q) = u(Q) ≥ min u for some Q ∈ V (0) , and the first inequality is trivial. Suppose, on the contrary, there exists Q ∈ V (1) \ V (0) at which v attains its minimum. By virtue of (3), since v(P ) ≥ v(Q) =: m for each point P close to Q, we have in fact v(P ) = v(Q) for each point P close to Q. In particular, v(P ) = m for all P ∈ V (1) \ V (0) , thus we can repeat the same argument at all P ∈ V (1) \ V (0) . But every point of V (0) is close to some point in V (1) \V (0) . Thus, v = m on V (1) , in particular, since v = u on V (0) , u is constant. We will now prove as a simple consequence of the maximum principle, that the oscillation of the harmonic extension of u on every 1-cell is estimated by a constant times the oscillation of u on V (0) . Let ! " (0) S := u ∈ RV : u(P1 ) = 0, ||u|| = 1 .

(6)

and, given a function f from a nonempty set A to R let OscA f = sup |f (x) − f (y)| = sup f − inf f . x,y∈A

A

A

Corollary 1. There exists γ ∈]0, 1[ such that for all u : V (0) → R we have sup OscVi (H(1;E) (u)) ≤ γ OscV (0) (u).

i=1,2,3

(0)

Proof. Suppose u ∈ RV , u nonconstant. Then, by the strong maximum principle, for every i = 1, 2, 3 and for every P ∈ V (0) we have min u ≤ V (0)

H(1;E) (u)(ψi (P )) ≤ max u and the inequalities are strict if P = Pi . Hence, V (0)

at least one of the two inequalities is strict for every P ∈ V (0) , and OscVi (H(1;E) (u)) < OscV (0) (u) for i = 1, 2, 3. Thus, putting sup OscVi (H(1;E) (u))

α(u) =

i=1,2,3

OscV (0) (u)

,

we have α(u) < 1. Since α is continuous it has a maximum γ < 1 on S. Since α(u + c) = α(u) for every c ∈ R, and α is positively 0-homogeneous, we have

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Roberto Peirone

α(u) = α for every nonconstant u ∈ RV

(0)

u − u(P ) 1 ≤γ ||u − u(P1 )||

.

Given a function u : V (0) → R, as hinted above, we want to extend it to a function v on V (∞) in the following way. Put v = u on V (0) , then we define v recursively on V (n+1) \ V (n) for n ≥ 0. For n = 0, v is the harmonic extension of u on V (1) . More generally, suppose v is already defined on V (n) 3 # and extend v on V (n+1) . Note that V (n+1) = ψi1 ,...,in (V (1) ). So, we i1 ,...,in =1

define v separately on every ψi1 ,...,in (V (1) ). Let v ψi1 ,...,in+1 (P ) = H(1;E) v ◦ ψi1 ,...,in ψin+1 (P )

(7) ∀ P ∈ V (0) , ∀ i1 , ..., in+1 = 1, 2, 3 , in other words, v ◦ ψi1 ,...,in+1 = H1;E v ◦ ψi1 ,...,in ◦ ψin+1 on V (0) . The right-hand side makes sense, as we have already defined v on V (n) . In some sense, by identifying a function on ψi1 ,...,in (V (1) ) with a function on V (1) , (7) defines v on ψi1 ,...,in (V (1) ) as the harmonic extension of the restriction of v on ψi1 ,...,in (V (0) ). The problem in defining v on V (n+1) by (7), is that we could suspect that a point Q can be represented as a point of V (n+1) in different ways, for example Q = ψi1 ,...,in+1 (P ) = ψi1 ,...,in+1 (P ),

(8)

and that the definition of v(Q) via (7) can depend on such a representation. We are now going to prove that this is not the case, so that (7) actually defines a function v. The geometrical reason for this is that we see that different (nth T ) copies can have as common points only their vertices, so that the points Q in V (n+1) \ V (n) can be represented in only one way as in (8). Such an argument can be considered as sufficiently persuasive. However, I will give a formal proof. We have to prove that if (8) holds then H(1;E) v ◦ ψi1 ,...,in ψin+1 (P ) = H(1;E) v ◦ ψi1 ,...,in ψin+1 (P ) .

(9)

If (i1 , ..., in ) = (i1 , ..., in ) then, as every map ψi is one-to-one, we must have by (8), ψin+1 (P ) = ψin+1 (P ), thus (9) holds. Suppose now (i1 , ..., in ) = (i1 , ..., in ). Then, it suffices to prove that ψin+1 (P ), ψin+1 (P ) ∈ V (0) , so that by definition of harmonic extension, we have H(1;E) v ◦ ψi1 ,...,in ψin+1 (P ) = v ψi1 ,...,in ψin+1 (P )

(10)


149

= v(Q) = H(1;E) v ◦ ψi1 ,...,in ψin+1 (P ) and (9) holds. In order to prove (10), we need the following lemma. Lemma 3. Suppose (i1 , ..., in ) = (i1 , ..., in ). Then, Ti1 ,...,in ∩ Ti1 ,...,in ⊆ Vi1 ,...,in ∩ Vi1 ,...,in . Proof. By the definition of ψi , we have ψi (T ) ∩ ψi (T ) ⊆ ψi (V (0) ) ∩ ψi (V (0) ) if i, i = 1, 2, 3, i = i . Thus, if im = im , and il = il for all l < m, in view of the trivial fact that every ψi maps T into itself, we have ψim ,...,in (T ) ∩ ψim ,...,in (T ) ⊆ ψim (T ) ∩ ψim (T ) ⊆ ψim (V (0) ) ∩ ψim (V (0) ). Clearly, every ψi maps T \ V (0) into itself, hence so does ψim+1 ,...,in . It follows that ψim+1 ,...,in (T ) ∩ V (0) ⊆ ψim+1 ,...,in (V (0) ) hence, using also the fact that every ψi is one-to-one, ψim ,...,in (T ) ∩ ψim ,...,in (T ) ⊆ ψim ,...,in (T ) ∩ ψim (V (0) ) = ψim ψim+1 ,...,in (T ) ∩ V (0) ⊆ ψim ,...,in (V (0) ) , ψi1 ,...,in (T ) ∩ ψi1 ,...,in (T ) = ψi1 ,...,im−1 ψim ,...,in (T ) ∩ ψi1 ,...,im−1 ψim ,...,in (T ) ⊆ ψi1 ,...,in (V (0) ) . The same argument is valid for i1 , ..., in in place of i1 , ..., in , thus we have proved the lemma. Now, since we have supposed (i1 , ..., in ) = (i1 , ..., in ), and the maps ψi are one-to-one, in view of (8) and Lemma 3, (10) follows at once, and thus (7) defines a function on V (∞) . We now want to prove that v is uniformly continuous on V (∞) , and hence it can be extended continuously on K. We will get this by proving that the oscillation of v on the n-cells tends to 0 as n tends to ∞. Let us give the following definition. For f : V (∞) → R, define sup OscV (∞) f . We have Osc,n f = i1 ,..,in =1,2,3

i1 ,...,in

Lemma 4. Osc,n (v) −→ 0. n→+∞

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Roberto Peirone

Proof. By (7) and Corollary 1, we have OscVi1 ,...,in+1 (v) ≤ γOscVi1 ,...,in (v) for all n ∈ N, i1 , ..., in+1 = 1, 2, 3, thus, for all i1 , ..., in = 1, 2, 3 we have OscVi1 ,...,in (v) ≤ γ n OscV (0) (u) . (∞)

Now, if P ∈ Vi1 ,...,in , we have P = ψi1 ,...,in ,in+1 ,...,ih (Q) for some Q ∈ V (0) and in+1 , ..., ih = 1, 2, 3. Thus, using (7) again and the maximum principle, a recursive argument yields min v ≤ v(P ) ≤ max v

Vi1 ,...,in

Vi1 ,...,in

and thus OscV (∞)

i1 ,...,in

(v) ≤ OscVi1 ,...,in (v) ≤ γ n OscV (0) (u) .

Since this holds for every i1 , ..., in , we have proved the lemma.

Corollary 2. v is uniformly continuous on V (∞) . Proof. Let ε > 0 be given. Let n be such that Osc,n (v) < 2ε . Since V ⊆ V (∞) ⊆ T , using Lemma 3 we get that for every i1 , ..., in , i1 , ..., in = 1, 2, 3, either Vi1 ,...,in (∞)

(∞)

and Vi ,...,i have nonempty intersection, or Ti1 ,...,in and Ti1 ,...,in are disjoint, n 1 and so, they being compact, they have a positive minimum distance. Let δ > 0 be less than the minimum of the distance of disjoint Ti1 ,...,in and (∞) (∞) Ti1 ,...,in . If P, P ∈ V (∞) and d(P, P ) < δ, we have P ∈ Vi1 ,...,in , P ∈ Vi ,...,i 1

n

for some i1 , ..., in , i1 , ..., in and there exists Q ∈ Vi1 ,...,in ∩ Vi ,...,i . Hence, n 1 |v(P ) − v(Q)| < 2ε and |v(P ) − v(Q)| < 2ε , thus |v(P ) − v(P )| < ε. (∞)

(∞)

We will call the continuous extension on K of v defined by (7), the harmonic extension of u on K, and we will denote it by H(∞;E) (u). I explicitly state the following lemma which is implicit in the proof of Lemma 4. Lemma 5. For every u : V (0) → R we have min u ≤ H(∞;E) (u) ≤ max u . Proof. Put v = H(∞;E) (u). By (7) and the maximum principle we have min v ≤

Vi1 ,...,in

min

Vi1 ,...,in+1

v≤

max

Vi1 ,...,in+1

v ≤ max v Vi1 ,...,in


151

so that, by a recursive argument we get min u = min v ≤ min v ≤ max v ≤ max v = max u , V (0)

Vi1 ,...,in

V (0)

Vi1 ,...,in

V (0)

V (0)

hence, min u ≤ v(P ) ≤ max u, for every P ∈ V (∞) , thus by continuity for every P ∈ K. The use of the harmonic extension is illustrated in the following theorem. Theorem 2. For every u ∈ RV

(0)

, we have

Σ (H(∞;E) (u)) = E(u) . E(∞)

Proof. Put H(∞;E) (u) =: v. Using also (5), we get Σ E(n+1) (v) =

=

ρn+1

i1 ,...,in =1

=

=

1 ρn 1 ρn

ρn+1

E(v ◦ ψi1 ,...,in+1 )

i1 ,...,in+1 =1

E H(1;E) (v ◦ ψi1 ,...,in ) ◦ ψin+1

3

3

1

3

1

in+1 =1 3

Σ H(1;E) (v ◦ ψi1 ,...,in ) E(1)

i1 ,...,in =1 3

Σ E(v ◦ ψi1 ,...,in ) = E(n) (v) ,

i1 ,...,in =1

Σ (H(∞;E) (u)) = E(u) for all n ∈ N. for all n ∈ N, thus E(n)

Σ among Note that by Theorem 2 we see that H(∞;E) (u) minimizes E(∞) (0) the functions defined on K which amount to u on V . We need the following simple lemma.

Lemma 6. We have 3

K=

ψi1 ,...,in (K) .

i1 ,...,in =1

Proof. We have V (∞) =

3 #

ψi1 ,...,in (V (∞) ). It now suffices to observe

i1 ,...,in =1

that K = V (∞) and that ψi1 ,...,in are continuous.

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Given a continuous function v on K, we can now construct for every m ∈ N, a function denoted by v(m;E) , defined on K, which is in some sense the harmonic extension of the restriction of v on V (m) . Namely, for every P ∈ K, put v(m;E) ψi1 ,...,im (P ) = H(∞;E) (v ◦ ψi1 ,...,im )(P ) . Thanks to Lemma 3, such a definition is correct, and the function v(m;E) is continuous, its restriction on every Ki1 ,...,im being continuous and the sets Ki1 ,...,im being closed subsets of K covering K. Moreover, using an argument like that in Theorem 2, we have Σ Σ (v(m;E) ) = E(m) (v) < +∞, E(∞)

so that v(m;E) has finite energy. We are going to prove that v(n;E) −→ v n→∞ uniformly. In fact, for every Q ∈ K let i1 , ..., in = 1, 2, 3, P ∈ K be such that Q = ψi1 ,...,in (P ). By the definition of v(n;E) and Lemma 5, we have v(n;E) (Q), v(Q) ∈ [ inf v, sup v]. Since Ki1 ,...,in

Ki1 ,...,in

1 (11) diamKi1 ,...,in ≤ ( )n diamK , 2 (we have in fact the equality in (11)) and v is uniformly continuous, it follows that v(n;E) −→ v uniformly, as claimed. In conclusion, n→∞

Theorem 3. The set of functions with finite energy is dense in C(K).

3 Dirichlet forms on finitely ramified fractals In this section we extend the construction described in the previous section from the case of the Gasket to the case of more general fractals. First, I describe other examples of fractals. The Vicsek set can be constructed analogously to the Gasket, by putting K0 to be a square, P1 , P2 , P3 , P4 its vertices, 5 # and P5 its centre, and ψi (x) = Pi + 13 (x − Pi ), and Kn+1 = ψi (Kn ), then i=1

using (1). The Sierpinski Carpet is also constructed starting from a square K0 , then splitting it into nine small squares of edge 13 , and considering the eight similarities carrying it into the small squares but the central one. The (Lindstrøm) Snowflake is obtained from a hexagon, and seven similarities, six of them having for fixed points the vertices of the hexagon, and the other one the centre of the hexagon. The tree-like Gasket is similar to the Gasket, but two of the three small triangles are disjoint. I finally also recall the celebrated Cantor set, where the initial set K0 is a segment-line, and there are two similarities having the end points of K0 for fixed points and of factor 13 . What could be a general definition of fractal? In all previous cases the set in Rν is obtained using an initial set and finitely many contractive (i.e, having a


153

factor < 1) similarities. By a slightly deeper investigation we realize that the initial set is not essential in this construction, as, for example in the case of the Gasket, we can start with V (0) instead of with T . In Figures 2 and 3, we describe the previous fractals, by depicting K1 .

øf

f Fig. 2. The Vicsek set and the Sierpinski Carpet

P2

P3

øf

P1

P2

P12

f

P1

P23

P4

P5

P3

Fig. 3. The Snowflake and the tree-like Gasket

What is the relationship between the similarities and the fractal obtained? The answer is: if ψ1 , ..., ψk are the similarities, then K is the unique k # nonempty compact subset of Rν such that K = ψi (K). In order to i=1

realize such a construction, we need some preliminary considerations. Fix ν = 1, 2, 3, ... in the following, and let K be the set of nonempty compact subsets of Rν . We! equip K with the so-called Hausdorff " distance, namely dH (C1 , C2 ) = sup d(x, C2 ) : x ∈ C1 , d(y, C1 ) : y ∈ C2 . Such a distance in some sense measures how much close are two sets. It is easy to prove that it is in fact a metric on K. We now prove that it is complete. Theorem 4. (K, dH ) is complete.

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Proof. Suppose Cn is a Cauchy sequence in K, and prove that it has a limit. There exists a subsequence of Cn , that we will call Dn such that dH (Dn , Dn+1 ) < 21n for all n. We will prove that Dn has a limit and this suffices to conclude the proof. Let " ! D = x ∈ Rν : ∃ xn ∈ Dn with x = lim xn . n→∞

We will prove that D is nonempty and compact and that D = lim Dn . In n→∞ order to achieve this, observe that for all n = 1, 2, 3, ... we have ∀ x ∈ Dn ∃ y ∈ D : d(x, y) ≤

1 2n−1

,

(12)

3 . (13) 2n In order to prove (12), note that by the definition of dH , there exists a sequence (x1 , ..., xm , ...) so that x = xn , and for all m = 1, 2, 3, ..., xm ∈ Dm and d(xm , xm+1 ) < 21m . Then xm is a Cauchy sequence and its limit y belongs 1 for all m ≥ n, to D. Clearly, by the triangular inequality, d(xn , xm ) < 2n−1 1 hence d(x, y) ≤ 2n−1 . Let us now prove (13). Let x ∈ D, ym ∈ Dm with x = lim ym , and let ∀ n ∀ x ∈ D ∃ xn ∈ Dn : d(x, xn ) ≤

m→∞

m > n be such that d(x, ym ) ≤ 21n . As above we find xn ∈ Dn such that 1 , and (13) follows at once. From (12) we see, in particular, d(xn , ym ) < 2n−1 that D is nonempty. Since Dn are bounded, in view of (13), so is D. In order to prove that D is closed, suppose xn ∈ D and x = lim xn , and prove that x ∈ D. Using (13) we find yn ∈ Dn : d(yn , xn ) ≤

n→∞

3 2n .

Hence, x = lim yn , and n→∞

x ∈ D. From (12) and (13) it immediately follows that D = lim Dn . n→∞

Suppose now ψi are contractive similarities in R , for i = 1, ..., k, with factors ri ∈]0, 1[, i.e. we have ν

||ψi (x) − ψi (y)|| = ri ||x − y|| ∀ x, y ∈ Rν . Let Φ : K → K be defined as Φ(C) =

k

ψi (C) .

i=1

We are searching for a fixed point of Φ. We have the following Theorem 5. Φ is a contraction of (K, dH ), hence it has a unique fixed point K, and moreover, Φn (A) −→ K for every A ∈ K. n→∞

Proof. Let r = max ri . Then, clearly, for every A, B ∈ K and for every i = 1, ..., k we have dH (ψi (A), ψi (B)) ≤ r dH (A, B). From the definition of dH it easily follows that Φ is a contraction with factor r.


155

In some important particular cases, we can also give a better characterization of Φn (A). For example, if T ∈ K is such that Φ(T ) ⊆ T , we easily see by recursion that Φn (T ) is a decreasing sequence of sets. It follows that ∞ ) Φn (T ) = K. In the case of the Gasket if T is the triangle, we see that n=0

Φn (T ) = Kn defined as in (1), thus we find again K =

∞ )

Tn . On the other

n=0 ψi if

ø = V ⊆ F , we hand, if we denote by F the set of the fixed points of see that V ⊆ Φ(V ), thus Φn (V ) =: V (n) is an increasing sequence, and we ∞ # V (n) = K. I stated this in the case of the Gasket. However, in easily get n=0

general, we do not require that V is the set of the fixed points of all ψi but of some ψi . This remark is important for the following. We will say that the set K as in Theorem 5 is the fractal (or self-similar set) generated by the set Ψ = {ψi : i = 1, ..., k} of contractive similarities. Note that a fractal can be generated by different sets of similarities. However, for the following we fix a set K as above with the associated similarities ψ1 , ..., ψk , and call Pi the fixed point of ψi for i = 1, ..., k, and put F = {Pi : i = 1, ..., k}. We have so explained what we mean by fractal. The next problem is how to construct an energy on it. We will do that by imitating the construction on the Gasket. However, as we will see, that kind of construction is not possible on every fractal, but we have in fact to require some additional hypotheses. Before analyzing what properties we need, we are going to give a more precise notion of the so far vague word energy. We require of course that an energy is a functional defined on C(K) which is nonnegative, quadratic, and which takes the value 0 on the constant functions. It also appears to be natural to assume that it takes the value 0 only on constant functions and that it is finite on a dense set of continuous functions. Moreover, it seems to be natural to require a property of compatibility with the fractal structure. The need of giving a notion of energy generalizing the Dirichlet integral has lead to the notion of Dirichlet form. Usually, a Dirichlet form is defined on an L2 space. Here, in order to avoid problems related to the construction of measures on the fractal, I prefer to define it on C(K), although this is, in some sense, less natural. Definition 1. We say that a functional E is a good Dirichlet form if it satisfies the following properties a) E is a quadratic form from C(K) to [0, +∞] in the sense that there exists a linear subspace Z of C(K) such that E(v) < +∞ if and only if v ∈ Z, and there exists Eˆ : Z × Z → R bilinear and symmetric such that ˆ v) ≥ 0 E(v) = E(v, for all v ∈ Z. b) E((v ∧ 1) ∨ 0) ≤ E(v) ∀ v ∈ C(K) (Markov property). c) E is lower semicontinuous (with respect to the L∞ topology).

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Roberto Peirone

d) E(v + c) = E(v) ∀ v ∈ C(K), ∀ c ∈ R, (thus, E is 0 on all constant functions). e) E is irreducible, i.e., E(v) = 0 ⇒ v constant. f ) There exists a set H ⊆ C(K) such that H is dense in C(K) with respect to the L∞ topology, and E(v) < +∞ for every v ∈ H. k g) ∃ ρ > 0 : E(v) = ρ1 E(v ◦ ψi ) i=1

Properties a), b), c) characterize the Dirichlet forms, d), e), f) are in some sense properties of regularity of a Dirichlet form, and g) is the self-similarity Σ constructed on the Gasket in Section 2 is a property. The functional E(∞) good Dirichlet form. Indeed, a) is trivial, b) easily follows from the analogous property of the function (x, y) → (x − y)2 (i.e. if s(t) = (t ∧ 1) ∨ 0, then Σ is the sup of the (s(x) − s(y))2 ≤ (x − y)2 ); c) follows from the fact that E(∞) Σ continuous functionals E(n) , d) is trivial, e) and f) have been proved in Section Σ . 2, and g) can be easily proved with ρ = 35 on the base of the definition of E(n) Σ We are now trying to imitate the construction of E(∞) on a general fractal. To do this, we have to restrict the class of fractals considered, requiring that analogous properties to those used in the construction on the Gasket hold in our fractals. Here, in order to simplify the presentation, we will not try to define the widest class of fractals suitable for these considerations, but we will restrict ourselves to consider a class of fractals which is at the same time simpler to define and sufficient to include the most usual cases. In the case of Gasket, we used the fact that ψi (T \ V ) ⊆ T \ V . Such a property was crucial in the construction of the harmonic extension on K of a function defined in V , which in turn allowed us to prove property f) of Def. 1. Also, we implicitly used the fact that the points Pj are different from each other, and that the points ψi (Pj ) are not in V (0) unless i = j. So, we are lead to require the following property which includes the previous two: Definition 2. We say that K has the (strong) nesting property if Pj1 = Pj2 when j1 = j2 , and there exists a set V = V (0) ⊆ F, V = {P1 , ..., PN }, 2 ≤ / V (0) if N ≤ k, so that, putting T = coV , we have ψi (T ) ⊆ T and ψi (x) ∈ x ∈ T \ {Pi }, and in addition ψi (T ) ∩ ψi (T ) = ψi (V ) ∩ ψi (V ) if i, i = 1, ..., k, i = i .

(14)

Note that we used (14) in Lemma 3. In the case of the Gasket, V and T are as in Section 2. In the absence of the nesting property, we cannot even Σ is finite for some nonconstant function. It follows that conclude that E(∞) (0) ψi (T \ V ) ⊆ T \ V (0) , for all i = 1, ..., k. Note that the nesting property implies in particular that every n-cell contains at most one point of V (0) for n > 0. In addition, Ph = ψi (Pj ),

h, j = 1, ..., N, i = 1, ..., k ⇒ i = j = h .


157

This means that every Ph ∈ V (0) belongs to precisely one 1-cell, i.e., Vh (:= ψh (V (0) )). The phrase strong nesting property is more appropriate, as usually nesting property has a weaker meaning, and properly is a weaker version of (14), but in the following we conventionally omit the word strongly. In some sense, in Def. 2, (14) is the most characterizing property, in which it distinguishes the most usual fractals. In fact, the Vicsek set, the tree-like Gasket and the Snowflake have the nesting property, but the Carpet has not, as (14) does not hold. In our examples we have N = k = 3 in the Gasket and in the tree-like Gasket, N = 4, k = 5 in the Vicsek set, N = 6, k = 7 in the Snowflake, so that we can in fact have N < k. Another property we have to require is suggested by the Cantor set. There, if we try to imitate the definition of M1 (E) we obtain 0, as, for every function defined on V , which in this case is the set of the end-points of the segment-line, it can be extended on V (1) to a function which is constant on each 1-cell. The reason is that the Cantor set is too much disconnected, not so in a topological sense but in the combinatorial sense that the 1-cells are disjoint. In order to give a precise notion of connectedness, we recall the following definitions about graphs. A graph is a pair (V, W ) where V is a nonempty set and W is a subset of the set of the subsets of V having precisely two elements. The elements of W will be called the edges of the graph. We will say that P, Q ∈ V are close (in (V, W )) if {P, Q} is an edge. We will say that P, Q ∈ V are connected (in (V, W )) if there exist n = 1, 2..., P1 , ..., Pn ∈ V such that P1 = P , Pn = Q, and {Pi , Pi+1 } ∈ W for i = 1, ..., n − 1. In such a case, we will say that (P1 , ..., Pn ) is a path that connects P and Q (in (V, W )) and has length n. We will say that (V, W ) is connected, if any two points in V are connected. When V is clear from the context we can identify the graph with W , and say for example that W is connected. Now define Ai1 ,...,in = ψi1 ,...,in (A) for A ⊆ K and i1 , ..., in = 1, ..., k, where k # ψi1 ,...,in is an abbreviation for ψi1 ◦ ... ◦ ψin , and put V (n) = Vi1 ,...,in , V (∞) =

∞ #

i1 ,..,in =1

V (n) as in the case of Gasket. We explicitly note that

n=0

V (0) ⊆ V (n) ⊆ V (∞) ⊆ K ⊆ T, and V (n) is increasing with respect to n. Then we say that our fractal is (1) connected " graph is connected: G1 = (V , W ) where W is the ! if the following set of ψi (Pj1 ), ψi (Pj2 ) with i = 1, ..., k, j1 , j2 = 1, ..., N , j1 = j2 . In other words, we require that for every P, Q ∈ V (1) there exists a finite sequence P1 , ..., Pm of points in V (1) such that, P = P1 , Q = Pm , and for every r = 1, ..., m − 1, Pr , Pr+1 belong to some 1-cell, or also, that any two points in V (1) can be connected by a path in V (1) whose edges are contained in some 1-cell (depending on the edge). It can be easily verified that the Vicsek set, the tree-like Gasket and the Snowflake are connected. We will say that a fractal is (strongly) finitely ramified if both it has the nesting property and it is connected. As for the nesting property, we will omit the word strongly for

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Roberto Peirone

sake of simplicity, although, the usual definition of finitely ramified fractals is more general. We will construct a good Dirichlet form on our fractal, which from now on will be always assumed to be finitely ramified. At first glance, in the construction in Section 2 we used other properties of the Gasket. For example, in the proof of the maximum principle, we used the fact, that every point in V (1) \ V (0) is close to a point close to a fixed point in V (0) . However, this tells in other words that a point in V (1) \ V (0) and a point in V (0) are connected in G1 by a path of length ≤ 3, and the connectedness can well replace this assumption. A more serious difficulty is that, when we proved that M1 (E) is a multiple of E we heavily used the very strong symmetry of the Gasket, for example in the Snowflake there is no reason that the coefficients of (u(P1 ) − u(P2 ))2 and of (u(P1 ) − u(P3 ))2 are the same, as P1 and P2 are connected in V (1) in an essentially different way from P1 and P3 . In order to avoid such a problem, we will consider more general quadratic forms on V (0) . Suppose K is a finitely ramified fractal with similarities ψ1 , ..., ψk . We (0) define D to be the set of functionals from RV into R satisfying the following property: there exist cj1 ,j2 (E)(= cj1 ,j2 ) ≥ 0 (j1 = j2 ) with cj1 ,j2 = cj2 ,j1 such that E(u) = cj1 ,j2 (u(Pj1 ) − u(Pj2 ))2 (15) 1≤j1 <j2 ≤N

for all u : V (0) → R. % to be the set of those E ∈ D which are irreducible, Moreover, we define D i.e., E(u) = 0 if and only if u is constant. Concerning the previous definitions, % However, in some cases, we will also need to we are only interested in D. consider the set D which has for example the advantage that it is in some sense a closed set. The difference with respect to the case of the Gasket is that in this case we consider forms with possibly different coefficients. Now, we define Sn (E) and Mn (E) as in Section 2, with the obvious variant that the indices in the sum in the definition of Sn (E) vary from 1 to k instead of from % 1 to 3. In order to imitate the construction in Section 2, we need an E ∈ D such that there exists ρ > 0 with M1 (E) = ρE, in other words, we have to % → D, % which as prove the existence of an eigenvector for the operator M1 : D we will see, is in general, nonlinear. Before discussing this problem, however, we need a more detailed analysis of the previous notions. For example, I have % into D. % First of all, we note that the not proved that M1 maps in fact D coefficients of E are unique (this enables us to use the notation cj1 ,j2 (E)). This is a consequence of the following remark. Remark 1. Given j1 , j2 = 1, ..., N , j1 = j2 , let uj1 ,j2 , vj1 ,j2 be the functions in (0) RV that take the value 0 at every P different from Pj1 , Pj2 , and such that uj1 ,j2 (Pj1 ) = uj1 ,j2 (Pj2 ) = vj1 ,j2 (Pj1 ) = 1, vj1 ,j2 (Pj2 ) = −1 .


159

Then, if E is defined as in (15) (not necessarily cj1 ,j2 ≥ 0), we must have cj1 ,j2 = 14 E(vj1 ,j2 ) − E(uj1 ,j2 ) , as can be easily verified. % into D. % In the case of We now want to prove that M1 maps in fact D Gasket, we evaluated precisely M1 (E). We did this by solving explicitly the system in (3). Clearly, this is not possible in the general case. However, it is not necessary to solve explicitly such a kind of system, but it is sufficient to prove that it has a unique solution. For in such a case it depends linearly on u and we can proceed as in Section 2. We need some further considerations on graphs in V (1) . Remark 2. It is easy to see that the irreducibility condition for E ∈ D amounts to the fact that the graph on V (0) , whose edges are the sets {Pj1 , Pj2 } such that cj1 ,j2 > 0, is connected. We will denote such a graph by G(E). Note that, roughly speaking, the irreducibility condition amounts to the fact that there are not too many coefficients equal to 0. For example, if N = 3, this means that at most one of the coefficients c1,2 , c1,3 , c2,3 is 0. % We call G1 (E) the graph in V (1) whose edges are the sets Fix E ∈ D. of the form {ψi (Pj1 ), ψi (Pj2 )} with i = 1, ..., k, j1 , j2 = 1, ..., N, j1 = j2 and cj1 ,j2 > 0. We say that two points Q and Q in V (1) close (resp. E-close) if they are close in G1 (resp. in G1 (E)). So, two distinct points are close if they lie in the same cell. We say that Q and Q are connected (resp. E-connected) if they are connected in G1 (resp. in G1 (E)), i.e., if there exists a path (Q1 , ..., Qm ) with Q1 , ..., Qm ∈ V (1) , m ≥ 1, and Q1 = Q, Qm = Q , and Qi and Qi+1 close (resp. E-close). In such a case we say that the path connects (resp. Econnects) Q to Q . We say that Q and Q are strongly connected (resp. strongly E-connected), and that the path strongly connects (resp. strongly E-connects) / V (0) . Note however, that by Q to Q , if we can also assume Q2 , ..., Qm−1 ∈ (1) are connected. our assumptions any two points in V Remark 3. It easily follows from Remark 2 that any two points that lie in the same 1-cell Vi are E-connected by a path (Q1 , ..., Qm ) with Qh ∈ Vi for each h. It follows that, if i > N , so that Vi contains no points of V (0) , then any two points in Vi are strongly E-connected. If, instead, i ≤ N , so that Pi is the unique point in V (0) ∩ Vi , then any point in Vi is strongly E-connected to Pi . Lemma 7. Every point Q ∈ V (1) is strongly E-connected with some point of V (0) . Proof. Fix Q ∈ V (0) . Since the fractal is connected, there exist Q1 , ..., Qm ∈ V (1) such that Q1 = Q, Qm = Q and for every h = 1, ..., m − 1, Qh and Qh+1 are close. Thus, by Remark 3, Qh and Qh+1 are E-connected, and therefore ˜ 1 , ..., Q ˜ l ) E-connecting Q and Q , let Q and Q are E-connected. In a path (Q (0) ˜ Q be the first element Qh ∈ V . Then Q and Q are strongly E-connected.

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Note that, unlike the case of the Gasket, in general we cannot conclude that Q is strongly E-connected with any point of V (0) . For example in the tree-like Gasket, if cj1 ,j3 = 0 the point P1 is not strongly E-connected with (n) P3 . As in Section 2, we define L(n, u) = {v ∈ RV : v = u on V (0) }. We can characterize the minimum point of M1 (E) in L(1, u) like in Section 2. (0)

Lemma 8. If u ∈ RV , then a function v ∈ L(1, u) satisfies M1 (E)(u) = S1 (E)(v) if and only if cj1 ,j2 v(P ) − v ψi (Pj2 ) = 0, ∀P ∈ V (1) \ V (0) (16) where the sum is extended over all i = 1, ..., k, j1 , j2 = 1, ..., N such that j1 = j2 and P = ψi (Pj1 ). (0)

Lemma 9. Suppose u ∈ RV and v ∈ L(1, u) satisfies M1 (E)(u) = S1 (E)(v). Suppose P ∈ V (1) \ V (0) and v(P ) = max v or v(P ) = min v. Then, we have v(Q) = v(P ) whenever Q ∈ V (1) is strongly E-connected to P . Proof. Suppose for example v(P ) = max v =: M . It follows from the hypothesis that v(P ) = v(P ) if P is E-close to P , as, in the contrary case, the left-hand side in (16) would be strictly positive. Now, let (Q1 , ..., Qm ) be a path strongly E-connecting P to Q. By recursion, v(Qr ) = M for all r = 1, ..., m, and in particular, v(Q) = M . (0)

Proposition 1. If u ∈ RV and v ∈ L(1, u) satisfies M1 (E)(u) = S1 (E)(v), then v satisfies the maximum principle: for every P ∈ V (1) min u ≤ v(P ) ≤ max u . V (0)

V (0)

Proof. We prove for example the second inequality. Suppose that v(P ) = max v =: M for some P ∈ V (1) \ V (0) . Using Lemma 7 and Lemma 9, we conclude that v(Q)(= u(Q)) = M for some Q ∈ V (0) . Thus, M ≤ max u. (0)

there exists a unique v ∈ L(1, u) satisfying Theorem 6. For every u ∈ RV M1 (E)(u) = S1 (E)(v), which we will denote by H(1;E) (u). Proof. Clearly, given v : V (1) → R, v ∈ L(1, u) and satisfies M1 (E)(u) = S1 (E)(v), if and only if v satisfies the equations in (16) and, in addition, the equations v(P ) = u(P ) for P ∈ V (0) . Thus, we have a linear system with #V (1) equations and #V (1) unknowns v(P ), P ∈ V (1) . Therefore, we have to prove that the corresponding homogeneous system has no nontrivial solutions. But the homogeneous system is the system corresponding to u = 0. By the maximum principle, the unique solution to such a system is the function v = 0. Clearly, H(1;E) has the following properties: a) H(1;E) is linear, thus continuous. (0) b) H(1;E) (u + c) = H(1;E) (u) + c for all u ∈ RV and c ∈ R.


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Remark 4. The strong maximum principle in general does not hold. For example, in the tree-like Gasket if E is the form defined before Lemma 8, and u(P1 ) = u(P2 ) = 0, then H(1;E) (u) = 0 on the 1-cell V1 . We need a further lemma. Lemma 10. If v : V (n) → R and Sn (E)(v) = 0, then v is constant on V (n) . Proof. We imitate the proof of Lemma 1. Observe that for every m ∈ N, if v : V (m+1) → R has a constant value ci on every ψi (V (m) ), i = 1, ..., k, then v is constant on V (m+1) . Indeed, in particular, v = ci on the 1-cell Vi . Since the fractal is connected, ci is independent of i, say ci = c for each i, and k # since V (m+1) = ψi (V (m) ), v = c on V (m+1) as claimed. It follows that, if i=1

v is nonconstant on V (n) , then v ◦ ψi1 ,i2 ,...,in is nonconstant on V (0) for some i1 , i2 , ..., in = 1, ..., k, thus Sn (E)(v) > 0. % Theorem 7. M1 (E) ∈ D. Proof. By an argument similar to that in Section 2, M1 (E) has a representation as in (15) for suitable coefficients cj1 ,j2 (= cj2 ,j1 ), j1 , j2 = 1, ..., N , j1 = j2 . It remains to prove a) M1 (E) is irreducible. b) cj1 ,j2 ≥ 0. Let us prove a). We have M1 (E)(u) = S1 (E)(H(1;E) (u)). If u is nonconstant, so is H(1;E) (u). Hence, in view of Lemma 10, M1 (E)(u) > 0. Let us prove b). Let j1 , j2 = 1, ..., N , j1 = j2 , and uj1 ,j2 , vj1 ,j2 be as in Remark 1, and let w = H(1;E) (vj1 ,j2 ). Since |w| ∈ L(1, uj1 ,j2 ), by the definition of M1 (E) we have M1 (E)(uj1 ,j2 ) ≤ S1 (E)(|w|) ≤ S1 (E)(w) = M1 (E)(vj1 ,j2 ) ,

(17)

the second inequality being an immediate consequence of the formula (|a| − |b|)2 ≤ (a − b)2 .

(18)

Now, b) follows from Remark 1. % and ρ > 0 such Next, we discuss the problem whether there exist E ∈ D that M1 (E) = ρE. In such a case we say that E is an eigenform and ρ is its eigenvalue. In many fractals an eigenform exists. Put E to be that form having all coefficients equal to 1. We have seen that in the Gasket E is an eigenform. Although the Vicsek set is less symmetric than the Gasket, it is not difficult to see that its symmetry properties are sufficient to guarantee that E is an eigenform (in some sense the kind of connection of two different points P, P ∈ V (0) through V (1) is independent of P, P ). In the tree-like Gasket a direct calculations shows that

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% with c1,3 = 0 is an eigenform with eigenvalue 1 . In fact, let every form E ∈ D 2 a = c1,2 , b = c2,3 . In the definition of M1 we minimize the functional a(u(P1 )−x)2 +a(u(P2 )−x)2 +b(u(P3 )−y)2 +b(u(P2 )−y)2 +b(z−x)2 +a(t−y)2 , where, v being the function to minimize, we put P12 = ψ1 (P2 ) = ψ2 (P1 ), P23 = ψ3 (P2 ) = ψ2 (P3 ), P4 = ψ1 (P3 ), P5 = ψ3 (P1 ), x = v(P12 ), y = v(P23 ), z = v(P4 ), t = v(P5 ). Clearly, for the minimum v we must have x = z, y = t, and we have to minimize separately the functions a(u(P1 )−x)2 +a(u(P2 )−x)2 with respect to x and b(u(P3 ) − y)2 + b(u(P2 ) − y)2 with respect to y. Clearly, the result is 1 1 1 a(u(P1 ) − u(P2 ))2 + b(u(P3 ) − u(P2 ))2 = E(u) . 2 2 2 For the Snowflake the problem is more complicated. Here, and in general, the map M1 , considered as a map from the coefficients of E to the coefficients of M1 (E), is a rational function. This, as the solution H(1;E) (u) of system (16) is given by the quotients of two determinants which are polynomial functions of the coefficients of E, and M1 (E)(u) = S1 (E)(H(1;E) (u)). So, we cannot expect that M1 is linear, and when ad hoc arguments such as symmetry do not work, the problem of the existence of an eigenform is not trivial. A result of Lindstrøm [9], states that in every nested fractal there exists an eigenform. A nested fractal is defined to be a fractal having properties similar to that of finite ramification, and further the following additional symmetry property: If j1 , j2 = 1, ..., N , j1 = j2 , then the symmetry φj1 ,j2 with respect to Wj1 ,j2 = {z : ||z − Pj1 || = ||z − Pj2 ||}, maps n-cells to n-cells for n ≥ 0 and any n-cell containing elements on both sides of Wj1 ,j2 is mapped to itself. See [9] for the precise definition. It is easy to see that the Gasket, the Vicsek set, and the Snowflake are nested, and the tree-like Gasket is not nested. In particular, on the Snowflake there exists an eigenform. It is not difficult to see that not all fractals have an eigenform. A rather general criterion for the existence of an eigenform also valid for non-nested fractals was given by C. Sabot in his doctoral thesis (1995) (cf. [18]). Improvements of the results of Sabot were given in [13]. From now on, we assume that in our fractal there exists an eigenform, and ˆ will denote a fixed eigenform. E ˆ the It is not difficult to see that ρ < 1 (see [18]). We can now repeat for E same construction as that for E on the Gasket and use the same definitions ˆ Σ , and of harmonic extension H of E ˆ (u). We get (∞;E) (n) ˆ Σ (v) is increasing with respect to n. If Theorem 8. For every v : K → R, E (n) ˆ Σ is a good Dirichlet form on K. ˆ Σ (v), then E ˆ Σ (v) = lim E we put E (∞) (n) (∞) n→∞


163

Proof. We imitate the proof in Section 2. We have to modify slightly the proof of Corollary 1. When there, we stated that the number γ is less than 1, we used the strong maximum principle, which, as seen in Remark 4, is no longer valid in this situation. However, a little modification of that argument, using substantially the maximum principle, is still valid. We have to prove that OscVi (v) < OscV (0) (u), whenever u is a nonconstant function from V (0) to R and v = H(1;E) ˆ (u), and i = 1, ..., k. To this aim, using the maximum principle, it is sufficient to prove that v cannot attain in the same 1-cell Vi both its maximum and its minimum on V (1) . Suppose on the contrary, max v = v(ψi (Pj1 )), min v = v(ψi (Pj2 )), j1 , j2 = 1, ..., N . In view of Remark ˆ 3, either i > N and ψi (Pj1 ) and ψi (Pj2 ) are strongly E-connected, or i ≤ N ˆ and ψi (Pj1 ) and ψi (Pj2 ) are both strongly E-connected to Pi . By Lemma 9, in the former case v(ψi (Pj1 )) = v(ψi (Pj2 )), in the latter case v(ψi (Pj1 )) = v(Pi ) = v(ψi (Pj2 )). Hence, max v = min v, and v, thus u, is constant, a contradiction. Note that any positive multiple of an eigenform is an eigenform as well. So, a natural question is: is the eigenform unique up to a multiplicative constant? The answer is yes in several cases, but not always. The first example of nonuniqueness was given by V.Metz in [10], where it was proved that in the Vicsek set, there are infinitely many eigenforms not multiple of each other. A simpler (but less symmetric) example is the tree-like Gasket (see [16]), as we previously saw. We are now going to prove that, independently of the uniqueness, the eigenvalue ρ does not depend on the eigenform, thus it is related to the fractal, and it can be called the renormalization factor of the fractal. We need some preparatory considerations. % the ratio Lemma 11. For any E, E ∈ D A(u) =

E (u) E(u)

has positive minimum, which I will denote by λ− (E, E ), and maximum, which (0) I will denote by λ+ (E, E ) on the set of nonconstant u ∈ RV . They are also, respectively, the minimum and the maximum of A on the set S, defined in (6). (0)

u−u(P1 ) ) for every nonconstant u ∈ RV , it suffices Proof. Since A(u) = A( ||u−u(P 1 )|| to observe that A is continuous, thus it has a maximum and a minimum on S.

Remark 5. We clearly have λ− (E, E )E ≤ E ≤ λ+ (E, E )E % for every E, E ∈ D.

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We now state some simple properties of Sn and Mn which we will use in the following without explicit mention. a) Sn (aE) = aSn (E), Mn (aE) = aMn (E), b) E ≤ E ⇒ Sn (E) ≤ Sn (E ), Mn (E) ≤ Mn (E ), % a > 0. By the expression E ≤ E we mean E(u) ≤ E (u) where E, E ∈ D, for all u ∈ RV and similarly for Sn (E) and Mn (E). Note that E ≤ E does not amount to cj1 ,j2 (E) ≤ cj1 ,j2 (E ) for every j1 , j2 = 1, ..., N , j1 = j2 ; for example, if N = 3 and c1,2 (E ) = c1,3 (E ) = 3, c2,3 (E ) = 0, and c1,2 (E) = c1,3 (E) = c2,3 (E) = 1, using the simple inequality (a+b)2 ≤ 2a2 +2b2 , we have E ≤ E . We remark that in the previous considerations we did not require that the forms are eigenforms. (0)

% then they have the same eigenLemma 12. If E and E are eigenforms in D, value. Proof. Suppose M1 (E) = ρE and M1 (E ) = ρ E and prove that ρ = ρ . In fact, we have aE ≤ E ≤ bE for some a, b > 0, by Remark 5. Using a) and n n b), we get M1n (E) = ρn E, M1n (E ) = ρ E , and aρn E ≤ ρ E ≤ bρn E, for every n ∈ N. Hence, if ρ = ρ , we have that either E or E is identically 0, % contrary to the assumption E, E ∈ D.

4 Main properties of renormalization and harmonic extension Σ In previous sections, we studied the convergence of E(n) when E is an eigenform. We now want to study the corresponding problem when E is any element % The first remark is that, as we know that the eigenvalue ρ is independent of D. of the eigenform, we can well define Σ = E(n)

Sn (E) ρn

% We are now going to study the following problem: Is the for any E ∈ D. Σ convergent in this more general case? In this case, unlike the sequence E(n) Σ case of an eigenform there is no reason that the sequence E(n) is increasing. So, the answer is less simple. In order to attack the problem, we need a more careful investigation of the renormalization operator Mn (E) and of the harmonic extension on V (n) , that, so far, we merely hinted. This is what we will do in this section. We will prove in particular that Mn+m (E) = Mm (Mn (E)). This will turn out to be a consequence of the nesting property, as, in order to minimize the sum of the copies of E on the (n + m)-cells, we can minimize for fixed values on V (m) independently on the different n-cells as they only overlap at points in V (m) , and then minimize the sum of such minima among


165

the possible values on V (m) . The statement of Theorem 9 is thus, in some sense intuitive. However, I will give a complete proof of it. In order to do this, we need two Lemmas. We will use the notation H(m,n;E) for H(m;Mn (E)) . Lemma 13. Suppose (i1 , ..., in ) = (i1 , ..., in ). Then, Ki1 ,...,in ∩ Ki1 ,...,in ⊆ Vi1 ,...,in ∩ Vi1 ,...,in . Proof. Note that K ⊆ T , hence Ki1 ,...,in ∩Ki1 ,...,in ⊆ Ti1 ,...,in ∩Ti1 ,...,in . Then, the proof is exactly the same as that in Lemma 3. Lemma 14. Suppose n ∈ N. Suppose P = ψi1 ,...,in (Q) = ψi1 ,...,in (Q ) with i1 , ..., in , i1 , ..., in = 1, ..., k and Q, Q ∈ K, and (i1 , ..., in , Q) = (i1 , ..., in , Q ). Then, Q, Q ∈ V (0) . Proof. We have (i1 , ..., in ) = (i1 , ..., in ), as in the contrary case, Q = Q . Hence, by Lemma 13, Q, Q ∈ V (0) . Theorem 9. % and u ∈ RV (0) the inf in the definition of i) For every n ∈ N and E ∈ D Mn (E) is in fact a minimum. It is unique and we will denote it by H(n;E) (u). % Moreover, Mn (E) ∈ D. % and u ∈ RV (0) , on V (0) we have ii) For every n, m ∈ N and E ∈ D H(n+m;E) (u) ◦ ψi1 ,...,in+m = H(n;E) H(m,n;E) (u) ◦ ψi1 ,...,im ◦ ψim+1 ,...,im+n . % we have iii) For every n, m ∈ N and E ∈ D, Mn+m (E) = Mm (Mn (E)) . Proof. Suppose i) holds for n and m (for all E and u), and prove that the function v : V (n+m) → R defined by v ψi1 ,...,im (Q) = H(n;E) H(m,n;E) (u) ◦ ψi1 ,...,im (Q) for Q ∈ V (n) , i1 , ..., im = 1, ..., k, satisfies v ∈ L(n + m, u), Sn+m (E)(v) ≥ Mm Mn (E) (u) ∀v ∈ L(n + m, u) , (19) and the equality holds if and only if v = v. Since we already know that i) holds for n = 1, a recursive argument then yields i), ii) and iii). First, note that the definition of v is correct, i.e., it does not depend on the representation of P ∈ V (n+m) as P = ψi1 ,...,im (Q) with i1 , ..., im = 1, ..., k, Q ∈ V (n) . If P ∈ V (m) , then P = ψi1 ,...,im (Q ) for some i1 , ..., im = 1, ..., k, Q ∈ V (0) . Thus, either (i1 , ..., im , Q) = (i1 , ..., im , Q ), then Q = Q ∈ V (0) , or (i1 , ..., im , Q) = (i1 , ..., im , Q ), and using Lemma 14 we have Q ∈ V (0) again. Hence, the / V (m) , definition of v gives v(P ) = H(m,n;E) (u)(P ). If, on the contrary, P ∈

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in view of Lemma 14 the above representation of P is unique. Next, we prove that v ∈ L(n + m, u). We just proved that v amounts to H(m,n;E) (u) on V (m) ⊇ V (0) , which in turn amounts to u on V (0) . Finally, if v ∈ L(n + m, u), we have

Sn+m (E)(v) =

k i1 ,...,im =1

=

k i1 ,...,im =1

k

E(v ◦ ψi1 ,...,im ◦ ψim+1 ,...,im+n )

im+1 ,...,im+n =1 k

Sn (E)(v ◦ ψi1 ,...,im ) ≥

Mn (E) v|V (m) ◦ ψi1 ,...,im

i1 ,...,im =1

= Sm (Mn (E)) v|V (m) ≥ Mm Mn (E) (u) .

Moreover, the first inequality is in fact an equality if and only if v ◦ ψi1 ,...,im = H(n;E) v|V (m) ◦ ψi1 ,...,im on V (n) for all i1 , ..., im = 1, ..., k, the second is an equality if and only if v|V (m) = H(m,n;E) (u), if and only if v ◦ ψi1 ,...,im = H(m,n;E) (u) ◦ ψi1 ,...,im on V (0) for all i1 , ..., im = 1, ..., k. Hence, the equality holds in (19) if and only if v = v. Corollary 3. We have Mn = M1n .

% we put M ˜ n (E) := E(n) := Mnn(E) . When E is Now, for every E ∈ D, ρ n an eigenform then Mn (E) = ρ E, so that E(n) = E. We easily deduce from Theorem 9 that E(n+m) = (E(n) )(m) % and n, m ∈ N, thus M ñ = M ˜ n . Clearly, E ∈ D % is an eigenform for all E ∈ D 1 ˜ 1 . In order to investigate the convergence if and only if it is a fixed point of M Σ we need some information on the convergence of E(n) . We will prove of E(n) Σ ˜ Σ where E ˜ is the limit is Γ -convergent to E in fact that the sequence E(n) (∞) of E(n) . The proof of the convergence of E(n) is the real problem in the proof Σ when E is not an eigenform. Clearly, when E is an of Γ -convergence of E(n) eigenform, E(n) = E −→ E. In order to prove the convergence of E(n) , it will n→∞

be useful to study the behaviour of H(n;E) (u) on the single n-cells, in other words, to study H(n;E) (u) ◦ ψi1 ,...,in . In this connection, another consequence of Theorem 9 is that we can split the map u → H(n;E) (u) ◦ ψi1 ,...,in into the composition of maps like u → H(1;E) (u) ◦ ψi . We need thus a reformulation of


H(1;E) (u) ◦ ψi which represents it as a function of u. So, define Ti;E : RV (0) RV by

167 (0)

→

Ti;E (u) = H(1;E) (u) ◦ ψi % u ∈ RV . Put also Ti,n;E := Ti;M (E) . In for every i = 1, ..., k; E ∈ D; n the previous definitions, we omit E when is clear from the context, and in such a case, we write Ti , Ti,n . The following properties can be easily verified. We have already discussed some of them in terms of properties of the map u → H(1;E) (u). (0)

(0) % We have Proposition 2. Let u ∈ RV , E ∈ D. i) Ti is linear. ii) Ti (u + c) = Ti (u) + c if c is constant. iii) Ti (u)(Pi ) = u(Pi ) for all i = 1, ..., N . iv) Ti;aE = Ti;E if a > 0.

Note that thanks to iii), for i = 1, ..., N , Ti maps the space Πi = {u ∈ RV

(0)

: u(Pi ) = 0}

into itself, and by ii) Ti is completely determined by its values on Πi RN −1 . Thus, Ti can be considered as a linear operator from RN −1 into itself. Another trivial remark is that, by iv), we have Ti,Mn (E) = Ti,E(n) . Putting n = 1 in Theorem 9 ii), a simple recursive argument on m yields Lemma 15. Let u ∈ RV

(0)

% Then , E ∈ D.

H(n;E) (u) ◦ ψi1 ,...,in = Tin ,0 ◦ Tin−1 ,1 ◦ · · · ◦ Ti1 ,n−1 (u) on V (0) , in particular, if E is an eigenform, we have H(n;E) (u) ◦ ψi1 ,...,in = Tin ◦ Tin−1 ◦ · · · ◦ Ti1 (u) on V (0) . % u ∈ RV (0) , Q ∈ V (n) . Then Corollary 4. Let E ∈ D, min u ≤ H(n;E) (u)(Q) ≤ max u . V (0)

V (0)

Proof. We have min u ≤ Ti;E (u) ≤ max u for every i = 1, ..., k. We conclude V (0)

by a recursive argument.

V (0)

% but I did not So far, I discussed about the convergence of sequences in D, specify by what a sense I mean the convergence. It will be better to consider, % as D is closed. We can define a a priori, the convergence on D instead of on D, norm || || on the linear space generated by D as ||E|| = sup |E(u)|. Note that u∈S

if ||E|| = 0, then as E is 2-homogeneous and satisfies E(u) = E(u − u(P1 )), (0) it follows E(u) = 0 for all u ∈ RV , so that || || is in fact a norm. We have:

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Roberto Peirone

Lemma 16. Given En , E ∈ D, the following properties are equivalent a) En −→ E pointwise, n→∞

b) En −→ E uniformly on the compact subsets of RV

(0)

n→∞

,

c) En −→ E in the norm || ||, n→∞

d) cj1 ,j2 (En ) −→ cj1 ,j2 (E) for all j1 , j2 = 1, ..., N , j1 = j2 . n→∞

Proof. Clearly, we have d) ⇒ b) ⇒ a). Also, a) ⇒ d) by Remark 1. Since the set S is compact, we have b) ⇒ c). Finally, as E(u) = ||u − (0) u−u(P1 ) 2 u(P1 )|| E ||u−u(P1 )|| for every nonconstant u ∈ RV , c) ⇒ a). % will be meant to be in one of So, the convergence in D (a fortiori in D) the four equivalent formulations in Lemma 16. Also, we will consider on D % the topology induced by such a convergence. In this way the main and on D % are continuous. functions we defined on D Lemma 17. We have % × RV (0) to RV (1) defined by (E, u) → H(1;E) (u), is i) The map from D continuous. % × RV (0) to R defined by (E, u) → M1 (E)(u) is conii) The map from D tinuous. % to D. % iii) Mn is continuous from D %×D % to ]0, +∞[. iv) λ+ and λ− are continuous from D % uh , u ∈ RV (0) , Eh −→ E, uh −→ u, and Proof. Prove i). Suppose Eh , E ∈ D, h→∞

h→∞

prove that H(1;Eh ) (uh ) −→ H(1;E) (u) .

(20)

h→∞

By the maximum principle we have min uh ≤ H(1;Eh ) (uh ) ≤ max uh . Thus, V (0)

V (0)

the functions H(1;Eh ) (uh ) are uniformly bounded and if (20) does not hold, we have H(1;Ehl ) (uhl ) −→ v l→∞

with v = H(1;E) (u), for some strictly increasing sequence of naturals hl . Thus, ii) easily v satisfies (16), and v = H(1;E) (u), contrary to our assumption. follows from i), in view of the formula M1 (E)(u) = S1 (E) H(1;E) (u) . iii) is an immediate consequence of ii) and of the formula Mn (E) = M1n (E). We now (0) prove iv). Suppose Eh −→ E, Eh −→ E . Given a nonconstant u ∈ RV , let h→∞

h→∞

α = E(u) and let h ∈ N be such that Eh (u) ≥ calculations, for every h ≥ h and u ∈ S we get

α 2

for h ≥ h. By simple


169

E (u) E (u) 2 h − ≤ 2 Eh (u)E(u) − Eh (u)E (u) Eh (u) E(u) α 2 ≤ 2 ||Eh || ||Eh − E|| + ||Eh || ||Eh − E || −→ 0 , h→∞ α so that iv) easily follows. % and E(n) −→ E , then E is an eigenform. Corollary 5. If E, E ∈ D

˜1 Proof. Since E = lim M n→∞ ˜ 1. point of M

n

n→∞

˜ 1 is continuous, then E is a fixed (E), and M

Lemma 17 and Corollary 5 will be used without mention in the following. % is given by the Another equivalent way of expressing the convergence in D following corollary. % then En −→ E ⇐⇒ λ± (E, En ) −→ 1. Corollary 6. Given En , E ∈ D, n→∞

n→∞

Proof. Part “if” follows from Remark 5. Part “only if” follows from Lemma 17 iv. In some sense, the functions λ± provide a way of measuring how much far % are. We are so lead to give the following definition. two different E, E ∈ D % Given E, E ∈ D let λ(E, E ) = ln(λ+ (E, E )) − ln(λ− (E, E )). λ is a particular case of Hilbert’s projective metric. For the theory of Hilbert’s projective metric see for example [15]. We have in fact, as can be % in the sense that it has all propeasily verified, that λ is a semimetric on D, erties of a metric but the property that the distance is 0 only if the two elements are the same. More precisely, we have λ(E, E ) = 0 if and only if % is an eigenform if and E is a (positive) multiple of E. In particular, E ∈ D only if λ(E, E(1) ) = 0. Also, we clearly have λ(aE, bE ) = λ(E, E ) for every % generated by a, b > 0. Thus, λ induces a metric on the projective space pr(D) % % D, that is the space of equivalence classes on D, mod the relation that iden% I will now describe some simple properties tifies E to aE for a > 0, E ∈ D. of λ. While the previous results, in Sections 1 to 4 can be found (possibly in a slightly different form) in textbooks on this topic, for example [6], the following properties of λ can be found in at least one among [12], [16], [18] (cf. also [11]). I here follow [16]. % we have Lemma 18. Given E, E ∈ D ) ≥ λ− (E, E ) i) λ− (M1 (E), M1 (E )) = λ− (E(1) , E(1) ) ≤ λ+ (E, E ) ii) λ+ (M1 (E), M1 (E )) = λ+ (E(1) , E(1) iii) λ(M1 (E), M1 (E )) = λ(E(1) , E(1) ) ≤ λ(E, E ).

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Proof. In i), ii), iii), the equalities are trivial, so we have to prove the inequalities. By Remark 5 we have λ− (E, E )M1 (E) ≤ M1 (E ) ≤ λ+ (E, E )M1 (E), hence, λ− (E, E ) ≤

M1 (E )(u) ≤ λ+ (E, E ) M1 (E)(u)

(0)

for every nonconstant u ∈ RV , and i) and ii) follow at once, and iii) is an immediate consequence of i) and ii). % in view of Lemma Now, since E(1) is a multiple of M1 (E) for E ∈ D, ˜ 1 is a weak contraction on pr(D) % with respect to λ. This fact 18, the map M ˜ n converges could be ñ = M suggests that a way for proving that the iterated M 1 to try to prove that the above considered map is in fact a strong contraction. Unfortunately, this is not true in general, as we saw that we can have two % different normalized eigenforms, which correspond to two different (on pr(D)) ˜ 1 . However, we can modify such an argument in this way: In fixed points of M ˜ since the order to prove that the sequence E(n) tends to some eigenform E, ˜ distance λ between E and E(n) is decreasing in view of the previous lemma, we can try to prove that it is not eventually constant, so that we can hope that it tends to 0. We will use this argument on the Gasket in next section. We are thus lead to investigate the cases in which in Lemma 18 iii) we have the equality. For the moment, however, let us discuss some simple consequences ˜ and real numbers a, b with b ≥ a > 0, of Lemma 18. Given an eigenform E let us put

!

% : aE ˜ ≤ E ≤ bE ˜ = E∈D

"

Ua,b,E˜ !

% : a ≤ λ− (E, ˜ E), λ+ (E, ˜ E) ≤ b = E∈D

"

ˆ is an eigenform and thus E ˆ(1) = E, ˆ it and write Ua,b for Ua,b,Eˆ . Since E ˜ 1 maps Ua,b into itself. Thus if immediately follows from Lemma 18 that M % belongs to Ua,b E ∈ Ua,b , then every E(n) lies in Ua,b . Note that every E ∈ D ˆ E), b = λ+ (E, ˆ E). Moreover, Ua,b is (sequentially) if, for example a = λ− (E, compact. We have in fact: Lemma 19. Every sequence in Ua,b has a subsequence convergent to some element of Ua,b . Proof. Let En be a sequence in Ua,b . Because of Remark 1, the coefficients ˆ j ,j ). Thus, En has a subsequence convergent cj1 ,j2 (En ) are estimated by 4b E(v 1 2 to some functional E and we immediately see that E ∈ D. Moreover, we ˆ ≤ E ≤ bE. ˆ Thus, if u ∈ RV (0) is nonconstant we have clearly have aE ˆ % Moreover, E ∈ Ua,b . E(u) ≥ aE(u) > 0, so that E ∈ D.


171

% there exists a strictly increasing sequence of Corollary 7. For every E ∈ D % naturals nh and E ∈ D such that E(nh ) −→ E . h→∞

The problem is thus to prove that all the sequence E(n) converges to the same limit E . The use of Ua,b is the point in which we need the existence of an eigenform. Lemma 19, in fact, states that in some sense the sequence E(n) is bounded with respect to λ. Another consequence of Lemma 18 is the following. % put Corollary 8. Given E, E ∈ D, ) λn = λn (E, E ) = λ(E(n) , E(n) ). λ±,n = λ±,n (E, E ) = λ± (E(n) , E(n)

Then i) λ+,n is decreasing and λ−,n is increasing, thus λn is decreasing ii) If we set λ± = lim λ±,n we have 0 < λ− ≤ λ+ < +∞. n→∞

Proof. Since E(n+1) = (E(n) )(1) , i) is an immediate consequence of Lemma ) in place of E . For ii), note 18, putting there (E(n) ) in place of E and (E(n) that 0 < λ−,0 ≤ λ−,n ≤ λ+,n ≤ λ+,0 < +∞. % and E is an eigenform and E(n ) −→ E for Corollary 9. If E, E ∈ D h h→∞

some strictly increasing sequence of naturals nh , then E(n) −→ E . n→∞

= Proof. By Corollary 6, λ± (E , E(nh ) ) −→ 1. On the other hand, since E(n) h→∞

E , there exists lim λ± (E , E(n) ) = 1. By Corollary 6 again, E(n) −→ E . n→∞ n→∞ We have just seen that λ+,n ≤ λ+,0 and λ−,n ≥ λ−,0 . In order to know ˜ n contracts the distance λ for some n, we need to whether the operator M know when such inequalities are in fact equalities. To this aim, we study the (0) set of functions in RV , which are in some sense extrema for the ratio EE . % put For E, E ∈ D ! " (0) (A± =)A± (E, E ) = u ∈ RV : E (u) = λ± (E, E )E(u) . By definition such sets include the constant functions. Since in some cases we need to use only nonconstant functions, put (A˜± =)A˜± (E, E ) to be the set of the functions in A± (E, E ) which are not constant. Put also (A±,n =)A±,n (E, E ) = A± (Mn (E), Mn (E )) = A± (E(n) , E(n) )) , (A˜±,n =)A˜±,n (E, E ) = A˜± (Mn (E), Mn (E )) = A˜± (E(n) , E(n) )) .

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Roberto Peirone

% Then Proposition 3. Let E, E ∈ D. ± i) A is closed. ii) u ∈ A± ⇒ c1 u + c2 ∈ A± ∀ c1 , c2 ∈ R. iii) E is a multiple of E ⇐⇒ A˜+ ∩ A˜− = ø. iv) A˜± = ø. Note that, if λ±,n = λ±,0 , then ! " (0) A±,n (E, E ) = u ∈ RV : Mn (E )(u) = λ±,0 Mn (E)(u) . Now, we can state the following result. Lemma 20. Let E and E and λ±,n be as in Corollary 8. If we have λ±,n = λ±,0 , then for every u ∈ A±,n we have H(n;E) (u) ◦ ψi1 ,...,in = H(n;E ) (u) ◦ ψi1 ,...,in ∈ A± for all i1 , ..., in = 1, ..., k. Proof. Let u ∈ A−,n . Then

Mn (E )(u) = Sn (E ) H(n;E ) (u) =

k

E H(n;E ) (u) ◦ ψi1 ,...,in ≥

i1 ,...,in =1

k

λ−,0 E H(n;E ) (u) ◦ ψi1 ,...,in

i1 ,...,in =1

= λ−,0 Sn (E) H(n;E ) (u) ≥ λ−,0 Mn (E)(u) = Mn (E )(u)

so that the two inequalities are in fact equalities. From the fact that the first inequality is an equality we deduce H(n;E ) (u) ◦ ψi1 ,...,in ∈ A− , and from the fact that the second inequality is an equality we deduce H(n;E) (u) ◦ ψi1 ,...,in = H(n;E ) (u) ◦ ψi1 ,...,in , for all i1 , ..., in = 1, ..., k. We have proved the Lemma for the case where ± is −. We can proceed similarly in the case where ± is +. Corollary 10. In the same hypotheses as in Lemma 20, for every m with 0 ≤ m ≤ n we have λ±,n = λ±,m = λ±,0 = λ±,n−m (Mm (E), Mm (E )) = λ±,0 (Mm (E), Mm (E )) , (21) and for every u ∈ A±,n H(n−m,m;E) (u) ◦ ψi1 ,...,in−m = H(n−m,m;E ) (u) ◦ ψi1 ,...,in−m ∈ A±,m

(22)

for all i1 , ..., in−m = 1, ..., k. Proof. From Corollary 8, (21) follows at once. Thus, (22) follows from Lemma 20 and the fact that u ∈ A± Mn−m Mm (E) , Mn−m Mm (E ) .


173

Note that the first equality in (22) for every i1 , ..., in−m = 1, ..., k, simply amounts to H(n−m,m;E) (u) = H(n−m,m;E ) (u). Roughly speaking, whenever we have λ±,n = λ±,0 , every function u ∈ A±,n produces, via its harmonic extension, functions in A±,m for all m ≤ n, at every (n−m)-cell, in particular, for m = 0, functions in A± , at every n-cell. % if λ(E(n) , E ) = λ(E, E ) then Remark 6. Note that, given E, E ∈ D, (n) λ± (E(n) , E(n) ) = λ± (E, E ).

5 Homogenization on the Gasket In this section we assume that the fractal is the Gasket. We first prove that for % the sequence E(n) converges, then we prove that the sequence every E ∈ D, Σ is Γ -convergent. We call this phenomenon homogenization. Analogous E(n) convergence results also hold for general fractals, but I prefer first to illustrate the process in the Gasket, because the proof is more natural and so can be better understood. A typical property of the Gasket (and of other, but not of all, fractals), is that the limit form of E(n) is a multiple of E. The proof of the convergence of E(n) on the Gasket presented in this section has not been published, but the idea is sketched in the introduction of [16]. The proof in general fractals presented in Section 6 follows more or less the approach of [16]. ) = λ± (E, E ) in the In Section 4 we studied what happens if λ± (E(n) , E(n) % Put now E = E, so that E(n) = E, and put E in place general case E, E ∈ D. of E . Since E is an eigenform, then (22) yields H(n−m;E) (u) ◦ ψi1 ,...,in−m ∈ A±,m if u ∈ A±,n . Hence, in view of Lemma 15, we get

% and λ± (E, E(n) ) = λ± (E, E) and u ∈ A± (E, E(n) ), Proposition 4. If E ∈ D n (u) ∈ A± (E, E) for every i = 1, 2, 3. then Ti;E Let Ti := Ti;E in the rest of this section. The plan of the proof of the convergence of E(n) is more or less the following. First, we will prove that % then we have if such a convergence does not take place for some E ∈ D, % λ± (E, E(n) ) = λ± (E, E) for some E ∈ D; next, we will use Prop. 4 to deduce that A˜+ and A˜− contain some eigenvectors of Ti , obtained as a limit of Tin and, finally we will prove that these eigenvectors are the same, so that by Prop. 3, E is a multiple of E, and thus E(n) = E. In order to prove the first step, I will use the following notation. By Corollary 8, λ(E, E(n) ) is decreasing % approaches in n, so that λ(E, E(n) ) ≤ λ(E, E) for each n. I will say that E ∈ D E if λ(E, E(n) ) < λ(E, E) for some, thus for sufficiently large, n. I will say that E goes to E if E(n) tends, as n → ∞, to a multiple of E. Then % which is not a multiple of E approaches E, then Lemma 21. If every E ∈ D % goes to E. every E ∈ D

174

Roberto Peirone

% By Corollary 7 there exist a strictly increasing sequence of Proof. Let E ∈ D. % such that E(n ) −→ E . For every m ∈ N we have naturals nh and E ∈ D h h→∞

) = λ E, lim (E(nh ) )(m) = λ(E, lim E(m+nh ) ) λ(E, E(m) h→∞

h→∞

= lim λ(E, E(m+nh ) ) = lim λ(E, E(n) ) , h→∞

n→∞

the last equality depending on the fact that the last limit exists. Thus, E does not approach E and by hypothesis, E = aE for some positive a, in particular it is an eigenform. By Corollary 9, E(n) −→ E . n→∞

In order to investigate the convergence of Tj , I first introduce the problem (0) by some preliminary considerations. If u ∈ RV we identify u with a vector of R3 by putting u = (u(P1 ), u(P2 ), u(P3 )). By this identification, Tj is a linear operator from R3 into itself. Suppose forexample j = 3. Then T3 maps R2 × {0} into itself. We have T3 (x, y, 0) = H(1,E) (x, y, 0)(ψ3 (Pi ))i=1,2,3 = ( 25 x + 15 y, 15 x + 25 y, 0) (see solution of (3)). Hence, putting e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), we get T3 (e1 ) = ( 25 , 15 , 0), T3 (e2 ) = ( 15 , 25 , 0). The really important point in these formulas is that T3 (e1 ) = (a, b, 0), T3 (e2 ) = (b, a, 0)

with a, b > 0 .

(23)

This can be deduced by the symmetry of P1 and P2 and by the strong maximum principle, with no explicit calculations. On the base of the form of T3 we see that T3 maps the positive cone D3 := {v ∈ R3 : v1 ≥ 0, v2 ≥ 0, v3 = 0, v = (0, 0, 0)} into the cone D3 := {v ∈ R3 : v1 > 0, v2 > 0, v3 = 0}. Suppose now that λ(E, E(n) ) = λ(E, E), and take un ∈ A˜± (E, E(n) ). Suppose for example it attains its minimum at P3 . By Prop. 3 ii), we can and do assume un (P3 ) = 0, so that un ∈ D3 . We have wn := T3n (un ) ∈ A˜± . Clearly wn −→ 0, but for our considerations it is specially interesting to know what n→∞

wn is the limit of the normalized vector vn := ||w (if it exists). Since T3 in some n || sense has the effect of mixing the first two components in a symmetric way, √ we can expect that vn −→ 22 (1, 1, 0); however, since the point un depends n→∞ on n, this cannot be simply proved on the base on the convergence of the iterated, but we need a sort of uniform convergence. We now come to give the precise statement of the convergence of Tjn . Let

Dj := {v ∈ R3 : vl ≥ 0 for l = j, vj = 0, v = (0, 0, 0)}, Dj := {v ∈ R3 : vl > 0 for l = j, vj = 0, } . √ √ √ √ √ n Let v 1 = 0, 22 , 22 , v 2 = 22 , 0, 22 , v 3 = 22 , 22 , 0 . Let T* j (v) = max v Tjn (v) i n ||Tjn (v)|| when Tj (v) = 0. Let q(v) = min v , where the maximum and the i minimum are taken over i = 1, 2, 3 with i = j, when v ∈ Dj . Then √


175

n Lemma 22. We have T* −→ v j , in the sense that j (Dj ) n→∞ sup q Tjn (v) −→ 1 . n→∞

v∈Dj

Proof. We suppose for example j = 3, the other cases being analogous. Define functions q1 , q2 on D3 by q1 (v) = vv21 , q2 (v) = vv12 . Thus q = max{q1 , q2 } . Note that q1 , q2 , and thus q, are continuous with values in ]0, +∞[ . By (23) we have for v ∈ D3 q1 (T3 (v)) =

1 5 v1 2 5 v1

+ 25 v2 2q1 (v) + 1 = q1 (v) + 2 + 15 v2

and a similar formula holds for q2 (T3 (v)). In other words, for i = 1, 2, we have qi (T3 (v)) = f (qi (v)) where f (t) =

2t + 1 . t+2

Note that f satisfies a) f (1) = 1. b) f is strictly increasing on ]0, +∞[. c) f (t) < t for every t > 1. We deduce q(T3 (v)) = f (q(v)). Also, for every t ≥ 1, f n (t) −→ 1 .

(24)

n→∞

Moreover, the convergence in (24) is uniform on every interval of the form [1, c] with c > 1, since, by b) we have 1 ≤ f n (t) ≤ f n (c) for t ∈ [1, c]. Now, q has a maximum M ≥ 1 on T3 (D3 ) as q is continuous and positively 0-homogeneous, and the set of unit vectors in D3 , so its image via T3 , is compact. Since q(T3n (v)) = f n (q(v)), we have sup q T3n (v)) −→ 1 .

v∈D3

n→∞

Corollary 11. Given a strictly increasing sequence nh of naturals, and vh ∈ nh T+ j (Dj ), we have vh −→ v j . h→∞

Proof. Let for example j = 3. By Lemma 22 we have q(vh ) −→ 1. If w is a h→∞

limit point of vh , then w ∈ D3 , as, in the contrary case, a subsequence of q(vh ) tends to +∞. Hence, by continuity, q(w) = 1, which implies w = v 3 . Since v 3 is the unique limit point of vh , then vh −→ v 3 . h→∞

% goes to E. Theorem 10. Every E ∈ D

176

Roberto Peirone

% that does not Proof. By Lemma 21 it suffices to prove that given E ∈ D + ˜ approach E, then E is a multiple of E. Take un ∈ A (E, E(n) ). Clearly, there exists a strictly increasing sequence of naturals nh such that all unh attain their minima at the same point Pj1 and their maxima at the same point Pj2 , with j1 = j2 . By Prop. 3 ii we also have wnh := unh − unh (Pj1 ) ∈ A˜+ (E, E(nh ) ), wn h := unh (Pj2 ) − unh ∈ A˜+ (E, E(nh ) ). Moreover, wnh ∈ Dj1 , nh (wn ) ∈ A+ by Prop. 4, and by Corollary w ∈ Dj . Thus, vn := T+ nh

2

h

j1

h

11, vnh −→ v j1 , hence v j1 ∈ A˜+ . By proceeding in a similar way, vn h := h→∞

nh + ˜+ T+ j2 (wnh ) ∈ A , and vnh −→ v j2 ∈ A . By the same argument there exist h→∞

j1 , j2 , with j1 = j2 such that v j1 , v j2 ∈ A− . Since we have only three points in V (0) there exists v j ∈ A˜+ ∩ A˜− , and, by Prop. 3 iii, E is a multiple of E. Remark 7. The argument in the proof of Theorem 10 heavily relies on the fact that N = 3. Aiming to possible extension of the proof to more general fractals, it could be useful to observe that a modification of that argument does not use N = 3, but uses the strong symmetry of the Gasket, which implies in particular that v j is symmetric with respect to the components different from j. We saw in proof of Theorem 10 that there exists j(= j(0)) = 1, 2, 3 such that v j ∈ A+ (E, E). On the other hand, clearly, E(n) does not approach E, so that we can apply the same argument to E(n) and there exists j(n) = 1, 2, 3 such that v j(n) ∈ A+ (E, E(n) ). We can now repeat the previous argument. Let nh be a strictly increasing sequence of naturals and let j = 1, 2, 3 be such nh that j(nh ) = j for all h. Then, by Prop. 4, T+ (v j ) ∈ A+ , and, since v j ∈ Dj , j by Corollary 11 nh (v j ) −→ v j ∈ A˜+ . T+ j h→∞ √ On the other hand, for j = j, v˜j := (1, 1, 1) − 2 v j ∈ Dj ∩ A+ , hence nh T+ vj ) −→ v j ∈ A˜+ . j (˜ h→∞

In conclusion, v j ∈ A˜+ for every j, and by the same argument, v j ∈ A˜− for every j, so that A˜+ ∩ A˜− = ø. Corollary 12. Every eigenform is a multiple of E. % is an eigenform, then E = Proof. It suffices to observe, that if E ∈ D E(n) −→ aE, for some a > 0. n→∞

Theorem 10 is interesting in itself. However, we will use it as a starting Σ % and E ˜ = lim E(n) . . Let now E ∈ D point to prove the Γ -convergence of E(n) n→∞ ˜ Σ = Γ (X−) lim E Σ , where X = RK with We are going to prove that E (∞)

n→+∞

(n)


177

˜ is an eigenform, E ˜ Σ is defined. The the metric L∞ . Note that, since E (∞) argument of the proof of Γ -convergence is due to S. Kozlov [7], who proved the Γ -convergence on the Gasket for forms E having two of the three coefficients equal, with respect to a topology which is different from L∞ , and induces a sort of Sobolev space on the Gasket. In any case the proof of [7] also works for general fractals once we know that the sequence E(n) is convergent. In these Notes, I follow an approach similar to that in [17], where, in fact, the problem is treated by a slightly more general point of view in the sense that the Γ -convergence with respect to different topologies is investigated there. Recall that, given a sequence of functionals Fn from a metric space X with values in R ∪ {+∞} ∪ {−∞}, Fn are said to be Γ (X−)-convergent to a functional F if for every x ∈ X i) there exist xn −→ x such that Fn (xn ) −→ F (x). n→+∞

n→+∞

ii) For every xn −→ x we have lim inf Fn (xn ) ≥ F (x). n→+∞

n→+∞

In order to obtain the Γ -convergence result, for every m, n ∈ N with n ≥ m and for every v : V (m) → R, we define v(n,m) : K → R to be the n-harmonic ˜ is a multiple of ˜ (or to E, which is the same as E extension with respect to E E) of v˜ : V (n) → R defined by v˜ ψi1 ,...,in (P ) = H(n−m;E) (v ◦ ψi1 ,...,im ) ◦ ψim+1 ,...,in (P ), P ∈ V (0) . The definition of v˜ is correct by the same argument as in Section 2, via Lemma 3. Then, we define v(n) = v(n,[ n2 ]) . We have Lemma 23. For every v ∈ C(K), v(n) −→ v with respect to L∞ . n→∞

Proof. For every Q ∈ K let i1 , ..., in = 1, ..., k, P ∈ K be such that Q = ψi1 ,...,in (P ). Then, by the definition of harmonic extension, v(n) (Q) = v(n) (ψi1 ,...,in (P )) = H(∞;E) ˜ ◦ ψi1 ,...,in (P ) ˜ v and by Lemma 5 we have v(n) (Q) ∈ [min v˜ ◦ ψi1 ,...,in , max v˜ ◦ ψi1 ,...,in ] . V (0)

V (0)

Moreover, by Corollary 4 we get v(n) (Q) ∈ min v ◦ ψi1 ,...,i[ n ] , max v ◦ ψi1 ,...,i[ n ] . V (0)

V (0)

2

2

In conclusion, v(n) (Q), v(Q) ∈

'

min

Ki1 ,...,i[ n ]

v,

2

hence |v(n) (Q) − v(Q)| ≤

max

Ki1 ,...,i[ n ]

v−

2

max

Ki1 ,...,i[ n ] 2

min

Ki1 ,...,i[ n ] 2

v, and by the uniform conti-

nuity of v on K and (11), v(n) −→ v uniformly. n→∞

& v ,

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Roberto Peirone

Lemma 24. For every v ∈ C(K) we have Σ Σ (v(n) ) = (E(n−[ n2 ]) )Σ E(n) ([ n ]) (v) ≤ E(n) (v) . 2

Proof. We first observe that k

Sn (E)(v(n) ) =

Sn−[ n2 ] (E)(v(n) ◦ ψi1 ,...,i[ n ] ) , 2

i1 ,..,i[ n ] =1 2

Sn−[ n2 ] (E)(v(n) ◦ ψi1 ,...,i[ n ] ) = Sn−[ n2 ] (E) H(n−[ n2 ];E) (v ◦ ψi1 ,...,i[ n ] ) = 2

2

Mn−[ n2 ] (E)(v ◦ ψi1 ,...,i[ n ] ) ≤ Sn−[ n2 ] (E)(v ◦ ψi1 ,...,i[ n ] ) , 2

k

Sn (E)(v) =

2

Sn−[ n2 ] (E)(v ◦ ψi1 ,...,i[ n ] ) . 2

i1 ,..,i[ n ] =1 2

It follows that Sn (E)(v(n) ) = S[ n2 ] Mn−[ n2 ] (E) (v) ≤ Sn (E)(v) . By dividing by ρn we get the Lemma.

Theorem 11. We have Σ ˜Σ Γ (X−) lim E(n) =E (∞) n→+∞

where X = C(K) with the metric L∞ . Proof. By Corollary 6, for every ε > 0 we have ˜ ≤ E(n) ≤ (1 + ε)E ˜ (1 − ε)E

(25)

for sufficiently large n. Now, given v ∈ X, by Lemma 23 v(n) −→ v in X. n→∞

Also, by Lemma 24 and (25), ' & Σ ˜Σ ˜Σ E(n) (v(n) ) = (E(n−[ n2 ]) )Σ ([ n ]) (v) ∈ (1 − ε)E([ n ]) (v), (1 + ε)E([ n ]) (v) 2

2

2

Σ ˜ Σ (v). It remains to prove (v(n) ) −→ E for sufficiently large n, so that E(n) n→∞ (∞) that given vn −→ v in X, then n→+∞

Σ ˜ Σ (v) , lim inf E(n) (vn ) ≥ E (∞) n→+∞


179

and it is clearly sufficient to show that for every m ∈ N Σ ˜ Σ (v) . lim inf E(n) (vn ) ≥ E (m) n→+∞

(26)

˜ Σ is increasing, if 0 < ε < 1, for sufficiently By Lemma 24 and (25) and since E (n) large n we have Σ ˜Σ ˜Σ (vn ) ≥ (E(n−[ n2 ]) )Σ E(n) ([ n ]) (vn ) ≥ (1 − ε)E([ n ]) (vn ) ≥ (1 − ε)E(m) (vn ) . 2

2

˜ Σ is continuous from X to R, we get Since vn −→ v and E (m) n→+∞

Σ ˜ Σ (v) . ˜ Σ (vn ) = (1 − ε)E lim inf E(n) (vn ) ≥ (1 − ε) lim inf E (m) (m) n→+∞

n→+∞

Since this holds for any ε ∈]0, 1[, (26) follows.

6 Homogenization on general fractals In this section we will extend the results of Section 5 to arbitrary finitely ramified fractals. The difficulty in imitating the proof in Section 5 consists in extending Theorem 10. Note however, that in general, we cannot expect that E(n) tends to a multiple of a fixed eigenform, as in such a case, we could prove as in Corollary 12, that the eigenform is unique up to a multiplicative constant, and this is no longer true in the general case. We remark that if % we have N = 2, then for every E ∈ D, 2 E(u) = c u(P1 ) − u(P2 ) ,

2 M1 (E)(u) = c u(P1 ) − u(P2 )

for some c, c > 0. Hence, E is in any case an eigenform, and the convergence of E(n) takes trivially place. Thus, we can assume N ≥ 3. What properties of the Gasket have we used in the proof of the convergence of E(n) ? We essentially used either N = 3 or the very strong symmetry of the Gasket. If the fractal is less symmetric, one could prove an analog of Lemma 22, but the limit vector is not symmetric with respect to the components different from j. Hence such a vector does not attain its maximum at all P ∈ V (0) , P = Pj , and the proof in Remark 7 does not work. Since in the present case, we want to prove the convergence of E(n) to an eigenform that, in case of nonuniqueness, may well depend on E, and we cannot guess what eigenform is the limit, we will not ˜ We ˜ E(n) ) tends to 0 for some specific eigenform E. try to prove that λ(E, will instead try to prove that λn := λ(E(n) , E(n+1) ) tends to 0. Note that E(n+1) = (E(1) )(n) , so that we can use Corollary 8 and Corollary 10 with E(1) in place of E . In particular, λn is decreasing. We say that E is λ-contracting if we have λn < λ0 for some, thus for sufficiently large, n. We have the following analog to Lemma 21.

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Roberto Peirone

% that is not an eigenform is λ-contracting, then Lemma 25. If every E ∈ D ˜ % ˜ such that E(n) −→ E. for every E ∈ D there exists an eigenform E n→∞

% By Corollary 7 there exists a strictly increasing sequence Proof. Let E ∈ D. % such that E(n ) −→ E . For every m ∈ N we have of naturals nh , and E ∈ D h h→∞

, E(m+1) ) = λ lim (E(nh ) )(m) , lim (E(nh ) )(m+1) = λ(E(m)

h→∞

h→∞

λ( lim E(m+nh ) , lim E(m+1+nh ) ) = lim λ(E(m+nh ) , E(m+1+nh ) ) h→∞

h→∞

h→∞

= lim λ(E(n) , E(n+1) ) . n→∞

Thus E is not λ-contracting, and by hypothesis it is an eigenform. By Corol lary 9, then E(n) −→ E . n→∞

Actually, in the argument of the proof of convergence on the Gasket, when we stated that a and b in (23) are positive, we also used the strong maximum principle. The strong maximum principle, in fact, will play an important role also in the argument in this section. This will lead us to restrict the class of fractals. The convergence result can be proved for all finitely ramified fractals, using a variant of the strong maximum principle, but the proof is considerably more technical and will be omitted. I will hint the idea at the end of this section. We saw in Prop. 1 and Remark 4 that H(1;E) (u) satisfies the maximum % We principle, but in general not the strong maximum principle when E ∈ D. will now see that, if cj1 ,j2 (E) > 0 ∀ j1 , j2 : j1 = j2 ,

(27)

then H(1;E) (u) even satisfies the strong maximum principle. Note that, if % satisfies (27), then E-close amounts to close and E-connected amounts E∈D to connected. % and (27) holds. Suppose u : V (0) → R, Proposition 5. Suppose that E ∈ D and v := H(1;E) (u) attains its maximum or its minimum at a point Q of V (1) \ V (0) . Then u is constant on V (0) . Proof. We first prove that any point Q ∈ V (1) \ V (0) is strongly E-connected to any point Q ∈ V (0) . The proof of Lemma 7 shows that there exists a path (Q1 , ..., Qm ) E-connecting Q to Q . We will prove that if such a path has minimum length among the paths E-connecting Q and Q , then / V (0) . Suppose on the contrary Qi = Ph ∈ V (0) with 1 < i < m. Q2 , ..., Qm−1 ∈ As Vh is the unique 1-cell containing Ph , then Qi−1 , Qi+1 ∈ Vh . Hence the path (Q1 , ..., Qi−1 , Qi+1 , ..., Qm ) E-connects Q and Q and has length m − 1, which contradicts our assumption. Now, from Lemma 9, if v attains its maximum or its minimum at Q, then u = v = v(Q) on V (0) .


181

We cannot apply directly Prop. 5 to prove the convergence of E(n) , as E does not necessarily satisfy (27), but we will now see that a relatively mild condition on the fractal implies that every M1 (E) satisfies (27). We first give a sufficient condition on E in order that M1 (E) satisfy (27). It can be easily proved that it is also a necessary condition. % and every two points in V (0) are strongly Lemma 26. Suppose that E ∈ D E-connected. Then M1 (E) satisfies (27). Proof. The proof is a variant of that of b) in Theorem 7. Let j1 , j2 , uj1 ,j2 , vj1 ,j2 , w be as in Theorem 7. It suffices to prove that M1 (E)(vj1 ,j2 ) > M1 (E)(uj1 ,j2 ), hence that in (17) at least one of the two inequalities is strict. If the second one is not strict, we have E(|w| ◦ ψi ) = E(w ◦ ψi ) for all i = 1, ..., k, and thus w cannot attain opposite signs at two E-close points, as the inequality in (18) is strict when a and b have opposite signs. Let now (Q1 , ..., Qm ) be a path strongly E-connecting Pj1 to Pj2 . Here m > 2 as Pj1 and Pj2 cannot lie in the same 1-cell. Since w(Q1 ) = w(Pj1 ) = vj1 ,j2 (Pj1 ) = 1 and w(Qm ) = w(Pj2 ) = vj1 ,j2 (Pj2 ) = −1, there exists h = 2, ..., m − 1 such that w(Qh ) = 0. As |w| does not satisfy Lemma 9 with P = Qh , then |w| = H(1;E) (uj1 ,j2 ), and the first inequality in (17) is strict. We now require that the fractal has a strong connectedness property. The argument in proof of Prop. 5 shows that any two points in V (0) are strongly connected. We require a slightly stronger condition. We say that the fractal is strongly connected if for every Q, Q ∈ V (0) there exist Q1 , ..., Qm ∈ V (1) such that Q1 = Q, Qm = Q and, for every h = 1, ..., m − 1 there exists i(h) = 1, ..., k such that Qh , Qh+1 ∈ Vi(h) , and in addition, i(h) > N for h = 2, ..., m − 2. Note that the last condition means that all cells but the first and the last contain no points of V (0) . It is easy to see that the Gasket, the Vicsek set and the Snowflake are strongly connected, while the tree-like Gasket is not. % Proposition 6. Suppose the fractal is strongly connected and E ∈ M1 (D). Then E satisfies (27). Thus H(1;E) (u) satisfies the strong maximum principle (0) for every u ∈ RV . % We will prove that every two points in V (0) Proof. Let E = M1 (E ), E ∈ D. are strongly E -connected. The proof imitates that of Lemma 7. Fix Pj1 , Pj2 ∈ V (0) . Let (Q1 , ..., Qm ) be a path as in definition of strongly connected fractal. For any h = 1, ..., m − 1, the points Qh and Qh+1 are E -connected by a path which remains in Vi(h) , in particular, for h = 2, ..., m − 2, it remains in V (1) \ V (0) . Since Q2 ∈ Vj1 (the unique 1-cell containing Pj1 ), by Remark 3 Q2 is strongly E -connected to Pj1 . For the same reason, Qm−1 is strongly E -connected to Pj2 . In conclusion, we have found a path that strongly E connects Pj1 and Pj2 .

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From now on, unless specified otherwise, we will assume that the fractal is strongly connected. (0)

as (u(P1 ), ..., u(PN )). In this way, u will be We will write any u ∈ RV identified to a vector in RN . We will denote by e1 , ..., eN the vectors of the canonical basis in RN . % Then for every i = 1, ..., k, j = 1, ..., N Corollary 13. Suppose E ∈ M1 (D). we have ⎧ ⎪ if i = j = h ⎨= 1 Ti;E (eh )(Pj ) = 0 if i = j = h ⎪ ⎩ ∈]0, 1[ if i = j . Proof. We have Ti;E (eh )(Pj ) = H(1;E) (eh )(ψi (Pj )). If i = j we have ψi (Pj ) ∈ V (1) \ V (0) , thus the conclusion follows from the strong maximum principle. If i = j the results is trivial. In the previous section, in order to prove the convergence of E(n) , we were lead, in view of Lemma 21, to investigate the implications of the equality λ± (E, E(n) ) = λ± (E, E). In the present case, in order to prove the convergence of E(n) , in view of Lemma 25, we have to investigate the implications of the equality λ(E, E(1) ) = λ(E(n) , E(n+1) ). Therefore, instead of using a version of Corollary 10 in the case in which E is an eigenform, we will use a version of Corollary 10 in the case E = E(1) . However, also in the present case, it will be sufficient to consider the case i1 = ... = im . We have % Let λ±,n = λ± (E(n) , E(n+1) ) for all n. If we Lemma 27. Suppose E ∈ D. have λ±,n = λ±,0 , then for all m with 0 ≤ m ≤ n we have λ±,m = λ±,0 and if u ∈ A±,n (E, E(1) ) Ti,m ◦ · · · ◦ Ti,n−1 (u) = Ti,m+1 ◦ · · · ◦ Ti,n (u) ∈ A±,m (E, E(1) ) . Proof. It suffices to put E = E(1) , i1 = · · · = im = i in Corollary 10, and to use Lemma 15, taking into account the definition of H(n−m,m;E) . We need an analog of Lemma 22 suitable for Lemma 27. Roughly speaking, we want to prove that the normalization of the composition of m linear operators, under suitable conditions, contracts the positive cone to a unique vector, which in this case may well be nonsymmetric. However, there is a more relevant difference with the case of Section 5, i.e., as suggested by Lemma 27 we need to consider the case where the operators are different. I will give a general theorem for operators in RM and then I will apply this to our case. Let M ≥ 2 be fixed for the following (in case M = 1, Theorem 12 is trivially satisfied with w = 1). We define " ! D = v ∈ RM : vi ≥ 0 ∀ i = 1, ..., M, v = 0 ,


183

! " D = v ∈ RM : vi > 0 ∀ i = 1, ..., M . ˜ = {v ∈ D : ||v|| = 1}, D ˜ = {v ∈ D : ||v|| = 1}. Let A be the set Let also D M M of linear operators T : R → R that map D into D or equivalently such that T (ei ) ∈ D for all i = 1, ..., M . Let N : RM \ {0} → RM be defined as v . We will now introduce on D a semimetric that we implicitly N (v) = ||v|| used in the proof of Lemma 22 with M = 2, and is standard in the PerronFrobenius Theory. Given u, v ∈ D put λ+ (u, v) =

max

i=1,...,M

λ (u, v) =

vi , ui

λ− (u, v) =

ln(λ+ (u, v))

−

min

i=1,...,M

vi , ui

ln(λ− (u, v)) .

% and in fact it is another case The definition of λ resembles that of λ on D, of Hilbert’s projective metric. It satisfies the same properties as λ: we have λ (u, v) = 0 if and only if v is a multiple of u, λ (au, bv) = λ (u, v) for every a, b > 0. Thus, λ induces a metric on the projective space pr(D ) generated by D , that is the space of equivalence classes on D , mod the relation that ˜ , it induces a metric, identifies v to av for a > 0, v ∈ D . In particular, on D which we denote by λ as well. It is not difficult to see that such a metric is equivalent to the euclidean one, in the sense that they generate the same topology. However, as we will not use that statement, I will not discuss it. We instead will use a weaker form in Theorem 12. We will denote the diameter ˜ with respect to λ by diamA. The use of λ is illustrated of a subset A of D by the following lemma. Lemma 28. If u, v ∈ D and T ∈ A, then we have λ (T (u), T (v)) ≤ λ (u, v) and the inequality is strict unless λ (u, v) = 0. Proof. For all i = 1, ..., M , we have M

T (v)i j=1 = M T (u)i

ai,j vj ai,j uj

j=1

where ai,j = T (ej ) i > 0. We have ai,j vj ≤ λ+ (u, v)ai,j uj and, if λ (u, v) = 0, i.e., v is not a multiple of u, the inequality is strict for at least one j. Thus T (v)i T (u)i ≤ λ+ (u, v) for every i = 1, ..., M . Hence, λ+ (T (u), T (v)) ≤ λ+ (u, v) and similarly, λ− (T (u), T (v)) ≥ λ− (u, v), thus λ (T (u), T (v)) ≤ λ (u, v), and the inequality is strict if λ (u, v) = 0. The following theorem is specially interesting in the case in which the operators coincide. This is a form of the well-known Perron-Frobenius Theorem. For information on the Perron-Frobenius theory see for example [19].

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Theorem 12. Let A be a compact subset of A. Let T1 , ..., Tn , ... be a sequence ˜ such that of operators in A . Then there exists w ∈ D N T1 ◦ ... ◦ Tn (D) −→ w n→∞

−→ 0. in the sense that sup ||w − v|| : v ∈ N T1 ◦ ... ◦ Tn (D)

n→∞

Proof. For every T ∈ A let T˜ = N ◦ T . Then, we have ˜ N T1 ◦ ... ◦ Tn (D) = T˜1 ◦ ... ◦ Tñ (D)

∀n ∈ N.

Thus, putting T˜m,n = T˜m ◦ ... ◦ Tñ when m ≤ n, we have to prove that there ˜ such that exists w ∈ D ! " ˜ −→ 0 . sup ||w − v|| : v ∈ T˜1,n (D) n→∞

(28)

˜ × A → D ˜ be defined as α(v, T ) = T˜(v). It is continuous, hence Let α : D ˜ , thus, M :=diam(B) < +∞. We have B :=Imα is a compact subset of D ˜ ⊆ B ⊆ D. ˜ It follows Tñ (D) ˜ ⊇ T˜1,2 (D) ˜ ⊇ ... ⊇ T˜1,n (D) ˜ ⊇ ... B ⊇ T˜1,1 (D)

(29)

We will prove that ˜ −→ 0 . diam(T˜1,n (D)) n→∞

(30)

For every η > 0 let Fη = {(u, v) ∈ B×B : λ (u, v) ≥ η} and let β : A ×Fη → R be defined as β(T, u, v) = λ (u, v) − λ T˜(u), T˜(v) . Since A × Fη is compact, β has a minimum mη on it which, by Lemma 28, is positive. We will now prove that, given n = 1, 2, 3, ... such that M − nmη < η, we have ˜ ≤ η. diamT˜1,n+1 (D)

(31)

˜ As Tñ+1 (u), Tñ+1 (v) ∈ B, As η is arbitrary, this implies (30). Let u, v ∈ D. we have λ (Tñ+1 (u), Tñ+1 (v)) ≤ M . Now, if λ (T˜m,n+1 (u), T˜m,n+1 (v)) < η for some m ≤ n + 1, by Lemma ˜ ˜ 28 we have λ (T1,n+1(u), T1,n+1 (v)) < η. In the contrary case, we have T˜m,n+1 (u), T˜m,n+1 (v) ∈ Fη for all m ≤ n+1. Hence, in view of the definition of mη , by a recursive argument we get λ (T˜1,n+1 (u), T˜1,n+1 (v)) ≤ M − nmη < η, and λ (T˜1,n+1 (u), T˜1,n+1 (v)) < η again. Thus, we have proved (31), hence


185

˜ is a decreasing sequence of nonempty compact also (30). Since by (29) T˜1,n (D) ∞ ) ˜ and, by (30) we have subsets of B, then there exists w ∈ T˜1,n (D), n=1

! " ˜ −→ 0 . sup λ (w, v) : v ∈ T˜1,n (D) n→∞

On the other hand, since λ generates on the euclidean compact set B a topology weaker than the euclidean one, λ is equivalent to the euclidean metric. Hence, we get (28), and the theorem is proved. Corollary 14. Under the hypotheses of Theorem 12, given a strictly increasing sequence nh of naturals and vh ∈ D we have N T1 ◦ ... ◦ Tnh (vh ) −→ w . h→∞

! % and let D = v ∈ RV (0) : Corollary 15. Let i = 1, ..., k be fixed, let E ∈ D " v(Pi ) = 0, v(Pi ) ≥ 0 ∀ i = i, v = 0 . Then for every m = 1, 2, ..., there exists (0) wm ∈ RV such that, for every strictly increasing sequence nh of naturals and for every vh ∈ D, we have N Ti,m ◦ · · · ◦ Ti,nh (vh ) −→ wm . h→∞

Proof. As previously seen, the operators Ti,n can be identified to linear operators from RN −1 into itself, as they map the (N −1)-dimensional linear space Πi into itself. By this identification, D corresponds to the previously defined D, with N − 1 in place of M . Given a, b > 0 such that Eρ ∈ Ua,b , for all m ≥ 1 we have E(m) = M1 ( Eρ )(m−1) ∈ M1 (Ua,b ), hence Ti,m = Ti;E(m) ∈ Ti;M1 (Ua,b ) . By Lemma 19 and Lemma 17, the set Ti;M1 (Ua,b ) is compact, and by Corollary 13 it is contained in A. We can now use Corollary 14. We are now ready to prove the main theorem in this section. % there exists an eigenform E ˜ ∈D % such that Theorem 13. For every E ∈ D ˜ E(n) −→ E. n→∞

Proof. By Lemma 25 it suffices to prove that, given any non λ-contracting % then E is an eigenform. For every natural n let un ∈ A˜+,n (E, E(1) ). E ∈ D, Let nh be a strictly increasing sequence of naturals and let i = 1, ..., N be such that min unh = unh (Pi ) for all h ∈ N. We can assume that unh (Pi ) = 0, so that unh ∈ D where D is defined as in Corollary 15. Since E is not λcontracting, by Remark 6, λ± (E(n) , E(n+1) ) = λ± (E, E(1) ) and we can apply Lemma 27, hence um,nh := N Ti,m,nh −1 (unh ) = N Ti,m+1,nh (unh ) ∈ A˜+,m (E, E(1) )

(32)

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Roberto Peirone

for all m ≥ 1, where Ti,m,n denotes Ti,m ◦ · · · ◦ Ti,n when m ≤ n. Thus, using (0) Corollary 15, there exists wm ∈ RV such that um,nh → wm = wm+1 as (0) h → +∞. Hence, there exists a unit vector w ∈ RV such that wm = w for all m ≥ 1. It follows from (32) that w ∈ A˜+,m (E, E(1) ) for all m ≥ 1, so that E(m+1) (w) = λ+ E(m) (w), where we put λ± = λ± (E, E(1) ), hence E(m) (w) = λm−1 E(1) (w), + for all m ≥ 1. By the same argument, there exists w ∈ A˜−,m (E, E(1) ) such that E(1) (w ) E(m) (w ) = λm−1 − for all m ≥ 1. But we know that for some a, b > 0 we have E(m) ∈ Ua,b for all naturals m. Thus, ˆ ˆ ) = bE(w E(m) (w) ≤ bE(w)

ˆ E(w) ≤ LE(m) (w ) , ˆ E(w )

ˆ

E(w) where L = ab E(w ˆ ) . This is possible only if λ− = λ+ , i.e., only if E is an eigenform.

We can now prove a Γ -convergence result in the exactly same way as in Section 5. Namely, % We have Theorem 14. Let E ∈ D. Σ ˜Σ =E Γ (X−) lim E(n) (∞) n→+∞

˜ = lim E(n) and X = C(K) with the metric L∞ . where E n→∞

As we previously saw, Theorem 13 holds, even if the fractal is not strongly connected. I now sketch the idea of the proof in this case. The problem is that, as previously hinted, the strong maximum principle does not hold. So, we cannot use Theorem 12 to deduce Corollary 15. However, we can prove the following variant of Theorem 12. For every B ⊆ {1, ..., N }, let ! " /B , ΠB = v ∈ RN : vi = 0 ∀i ∈ ! " DB = v ∈ ΠB : vi ≥ 0 ∀i ∈ B, v = 0 , ! " DB = v ∈ ΠB : vi > 0 ∀i ∈ B . Theorem 15. Suppose T1 , .., Tn , ..., T∞ are linear maps from RN to RN , and there exists B ⊆ {1, ..., N } such that i) Tn maps ΠB into itself for every n ∈ N ∪ {∞} ∪ {0} for every n ∈ N ∪ {∞} ii) Tn maps DB into DB


187

iii) There exists a strictly increasing sequence of indices nh such that Tnh −→ T∞ , and DB ∩ KerTnh = DB ∩ KerT∞ for every h ∈ N. h→∞

Then there exists v ∈ D such that, for every strictly increasing sequence of naturals nh , for every vh ∈ D we have N (T1 ◦ ... ◦ Tnh (vh )) −→ v . h→∞

In other words, the condition that Tn maps D into D is replaced by the condition that Tn maps D into D ∪ {0}, and the part of D mapped into 0 is constant on a suitable compact set containing a subsequence. Theorem 15 fits in the situation of fractals which are not strongly connected. In fact, the positivity of Ti,n;E (ej )(Pj ) is related to the graph G(E(n) ). Here we identify V (0) with {1, ..., N }, and accordingly we interpret ΠB , DB , DB . Usually, G(E(n) ) can change at any step, and it need not satisfy the hypotheses of Theorem 15 if we do not require some additional hypothesis. However, this is the case if E is not λ-contracting. More precisely, we have: % Lemma 29. There exists n3 ≥ 1 such that, if h ≥ n1 + n2 + n3 , E ∈ D, u ∈ A˜±,h (E, E(1) ) with λ±,h (E, E(1) ) = λ± (E, E(1) ), then there exist m with 0 ≤ m ≤ n3 , i1 , ..., im = 1, ..., k, j = 1, ..., N , B ⊆ V (0) \ {Pj }, a = 1, −1, such that i) a Hm,h−m (u) ◦ ψi1 ,...,im − c ∈ D where c = Hm,h−m (u) ◦ ψi1 ,...,im (Pj ). ii) Tn := Tj,n+n1 , satisfies i) and ii) of Theorem 15. (0)

Of course, we here identify RV with RN , and, by this identification, v(Pj ) corresponds to vj . In other words, while in the case of strong maximum principle we can start with a function u ∈ D, which is mapped into D by one operator Tj,n , in the present case i) states that we have to use m operators before mapping u into D . After doing this, on the base of ii), we are able to apply Theorem 15. We have, in fact, still to verify iii) of Theorem 15, which however follows from a compactness argument. In fact, such a compactness argument holds for E in place of E where E is a suitable limit point of E(n) , namely a limit point for which the graph G(E ) has the minimal number of elements. Lemma 29 plays in this more general case the role that Corollary 13 played in the case of strongly connected fractals. The proof of Theorem 15 is a not too complicated variant of the proof of Theorem 12. On the contrary, the proof of Lemma 29 is the most delicate step in the proof of convergence of E(n) . At this moment, I do not know any simplification of that proof. Using Theorem 15 and Lemma 29 it is possible to prove again Theorem 13, and consequently, Theorem 14. In the proof of Theorem 13, however, there are some additional technical points with respect to the case of strongly connected fractals. Complete details can be found in [16], Section 4. There, Theorem 13 was proved in the more general setting of combinatorial fractal structures. The idea of investigating combinatorial fractal structures, firstly considered in [3], is motivated by the fact that Sn (E), Mn (E) and so on only depend on the graph G1 (E) and not on the geometry of the fractal. Theorem 13 was proved in [16] in its full generality, i.e., fractal structures having an

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eigenform. For previous proofs in particular cases, see for example [7] and [11], Example 8.8 and references therein for the case of Gasket, [12] for nested fractals with coefficients only depending on the distance; in [14] the result in strongly symmetric fractals (e.g., the Gasket) was announced without proof.

References 1. M.T. Barlow, Diffusions on Fractals, in Lectures on Probability Theory and Statistics, Lecture Notes in Mathematics 1690, Springer, Berlin, 1998. 2. M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math. 19, de Gruyter, Berlin, 1994. 3. K. Hattori, T. Hattori, H. Watanabe, Gaussian Field Theories on General Networks and the Spectral Dimension, Progr. Theor. Phys. Suppl. 92 (1987), 108143. 4. J.E. Hutchinson, Fractals and Self Similarity, Indiana Univ. Math. J. 30 (1981), 713-747. 5. J. Kigami, Harmonic Calculus on p.c.f. Self-similar Sets, Trans. Amer. Math. Soc. 335 (1993), 721-755. 6. J. Kigami, Analysis on fractals, Cambridge Tracts in Mathematics 143. Cambridge University Press, Cambridge, 2001. 7. S.M. Kozlov, Harmonization and Homogenization on Fractals, Comm. Math. Phys. 153 (1993), 339-357. 8. S. Kusuoka, Diffusion Processes on Nested Fractals in Statistical Mechanics and Fractals (S. Kusuoka and R.L. Dobrushin eds.), Lect. Notes Math. 1567, Springer-Verlag, Berlin, 1993. 9. T. Lindstrøm, Brownian Motion on Nested Fractals, Mem. Amer. Math. Soc. 420, 1990. 10. V. Metz, How many Diffusions Exist on the Vicsek Snowflake?, Acta Appl. Math. 32 (1993), 227-241. 11. V. Metz, Renormalization of Finitely Ramified Fractals, Proc. Roy. Soc. Edinburgh A 125 (1995), 1085-1104. 12. V. Metz, Renormalization Contracts on Nested Fractals, J. Reine Angew. Math. 480 (1996), 161-175. 13. V. Metz, The short-cut test, J. Functional Anal. 220 (2005), 118-156 14. S. Mortola, R. Peirone, Homogenization of Quadratic Forms on Fractals, Rend. Sem. Mat. Fis. Milano 64 (1994), 117-128. 15. R.D. Nussbaum, Hilbert’s Projective Metric and Iterated Nonlinear Maps, Mem. Amer. Math. Soc. 391, 1988. 16. R. Peirone, Convergence and Uniqueness Problems for Dirichlet Forms on Fractals, Boll. U.M.I. (8) 3-B (2000), 431-460. 17. R. Peirone, Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals, Potential Anal., 21 (2004), 289-309. 18. C. Sabot, Existence and Uniqueness of Diffusions on Finitely Ramified Self´ Similar Fractals, Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), 605-673. 19. E. Seneta, Non-negative Matrices and Markov Chains, Springer-Verlag, New York, 1981. 20. R. Strichartz, Differential Equations on Fractals: a Tutorial, Princeton University Press, to appear.

Homogenization in perforated domains Gregory A. Chechkin Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University Moscow 119992, Russia e-mail: [email protected]

Introduction The main goal of these lectures is to show the effect which one can discover studying problems in perforated domains. In different situations the effective behavior of such domains is described by the homogenized equation involving a “term etrange” (additional nontrivial potential). First this effect was studied in [21]. The proofs in [21] are complicated but the authors considered the nonperiodic general case. They used the method of potentials for solving boundary value problems. The notion of “term etrange” was introduced in the famous paper [13], where the authors considered the periodic case and suggested an elegant proof of the homogenization result. Now one can find different papers where other methods were used for studying such problems. See for instance [6], [7], [8] [14], [26], [27], [31]. We want to present two main types of such situations. The first one is connected with the investigation of a boundary value problem in perforated domain with low concentration of holes and Dirichlet boundary conditions on the holes; in the second situation we study a boundary value problem in a perforated domain with Fourier boundary conditions posed on the boundary of the holes. We study the problem in the second case assuming that either the coefficients in the boundary condition on the perforation, are small or the mean values of the coefficients are small (see also [6]). At present, there is a rich collection of results on asymptotic analysis of problems in perforated domains, among them the homogenization results obtained for periodic, almost-periodic, and random structures. A detailed bibliography can be found, for example, in [11], [17], [19], [20], [21], [25], [30]. In particular, the problems involving Fourier–type boundary conditions at the boundary of the holes, were studied in [5], [7], [8] [10], [16], [27], [28], [31], [32] and [33]. In the special case, when the problems under consideration are dissipative (which is ensured by proper signs of the coefficients in the boundary conditions), the weak convergence of solutions was investigated in [14], [15], [16].

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Gregory A. Chechkin

Applying the method of compensated compactness [12], [22] and the twoscale convergence method [1], [24] (see also [23], where the method of two-scale convergence was adapted for perforated domains), one can obtain a homogenized problem and prove the convergence of the solutions of the original problem towards the solution of the homogenized problem. However, these methods do not allow to estimate the rate of convergence. In the paper we use the Bakhvalov technique of asymptotic expansions [2], [3] (see also [4]) which, while requires an additional regularity of solutions, makes it possible to construct correctors and obtain error estimates. We show how it works in the second part of this material. For simplicity in the second section we assume that the perforation is periodic and does not intersect the outer boundary of the domain. However, the approach used in the second part could be applied to the structures with more general microscopic geometry, for instance, to domains with locally periodic perforation. A special attention will be paid to the coerciveness of operators associated with the boundary value problems studied; it turns out that it is a nontrivial issue. It will be shown that it is sufficient to verify the coerciveness of the limit operator (see for details [6] and [29]).

1 Appearence of a “term etrange”. Dirichlet problems in domains with low concentration of holes 1.1 Basic notation and setting of the problem Suppose that Ω is a domain in R with smooth boundary, Q = {x : |x| < 1} 3 is the unit ball in R centered in the origin. Denote Ω ε = Ω\ (ε3 Q + 2εz), 3

z∈Z

3

i.e. Ω ε is a domain with holes of radius ε3 periodically situated with period 2ε (see Figure 1). Consider the following boundary value problems: in Ωε, ∆uε = f (x) ◦ (1) uε ∈ H 1 (Ω ε ),

(∆ + µ)u = f (x) ◦

u ∈ H (Ω), µ = 1

in − π2 ,

Ω,

(2)

An interesting effect in this case is connected with the small deviation of solution uε from the solution u for sufficiently small parameter ε, i.e. Problem (2) is the homogenized problem for the original problem (1). Note that

Homogenization in perforated domains

191

Fig. 1. Perforated domain

the main symbol of the operator of the homogenized problem has the same form as the main symbol of the operator of the original problem. In addition it should be noted that in the equation of the homogenized problem we have the “potential” (“term etrange”). 1.2 The homogenization theorem In this section we formulate and prove the basic Theorem. Denote by wε the following 2ε-periodic in x function ⎧ 3 ⎪ ε Q + 2εz , 0, if x ∈ ⎪ ⎪ ⎪ 3 ⎪ ⎪ z∈Z ⎪ ⎪ 3 ⎨ 1, if x∈R \ (εQ + 2εz), (3) wε (x) = 3 z∈ Z ⎪ 1 ⎪ 1 ⎪ ⎪ − 3 ⎪ ⎪ r ε , if x∈ εQ\ε3 Q + 2εz . ⎪ ⎪ ⎩ 1ε − ε13 3 z∈Z

Theorem 1. For solutions uε and u to problems (1) and (2) respectively the following estimates uε − wε u H 1 (Ω ε ) ≤ Cε f C α (Ω) ,

uε − u L2 (Ω ε ) ≤ Cε f C α (Ω)

are valid, where wε (x) is defined in (3) and C, α = const > 0, C does not depend on ε and f (x). Proof. Using the explicit formula (3) for wε one can check that if ε → 0, then wε − 1 L2 (Ω ε ) → 0, ∇wε L2 (Ω ε ) → 0 and the following estimates wε − 1 L2 (Ω ε ) ≤ C1 ε2 ,

(4)

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Gregory A. Chechkin

∇wε L2 (Ω ε ) ≤ C2 ε

(5)

hold true, where C1 and C2 do not depend on ε. Moreover, using the standard definition of the norm in the space H −1 (Ω ε ) and formula (3), one can prove that for any smooth function u the estimate (∆wε − µ)u H −1 (Ω ε ) ≤ C3 ε u

(6)

◦

H 1 (Ω) ◦

is valid, where the constant C3 > 0 is independent of ε and u ∈ H 1 (Ω). Using the integral identities for Problems (1) and (2), we estimate the following integral: ∇(uε − wε u)∇ψ dx Ωε

= ∇uε ∇ψ dx − u∇wε ∇ψ dx − wε ∇u∇ψ dx Ωε

Ωε

Ωε

= − f ψ dx − u∇wε ∇ψ dx − ∇u∇ψ dx − (wε − 1)∇u∇ψ dx Ωε

≤

Ωε

Ω

Ω

f ψ dx + u∇wε ∇ψ dx + µuψ dx + (wε − 1)∇u∇ψ dx Ωε

Ω\Ω ε

Ω

Ω

= I1 + I2 + I3 . It is easy to see that I1 ≤ C4 ε f C α (Ω) ψ H 1 (Ω ε ) . Integrating by parts, keeping in mind inequalities (5), (6) and the following Schauder estimate (see for instance, [18]) for the solution to Problem (2): u C α+2 (Ω) ≤ C5 f C α (Ω) , where α > 0, C5 > 0 are constants and C5 > 0 does not depend on f (x), one can estimate I2 as follows: I2 ≤ C6 ε f C α (Ω) ψ H 1 (Ω ε ) . By (4) we obtain I3 ≤ C7 ε f C α (Ω) ψ H 1 (Ω ε ) and finally ∇(uε − wε u)∇ψ dx ≤ C8 ε f C α (Ω) ψ H 1 (Ω ε ) Ωε

(7)


193

Substituting ψ = uε − wε u in (7) and using the Friedrichs inequality ψ 2 dx ≤ C9 |∇ψ|2 dx, Ωε

Ωε

we get the first inequality of the Theorem. The second inequality of Theorem 1 is a consequence of the first one if we keep in mind the definition of wε . Now we complete the proof of the Theorem.

2 Homogenization problems in perforated domain with oscillating Fourier boundary conditions 2.1 Basic notation Let Ω be a bounded domain in R , d ≥ 2, with a smooth boundary. Consider the sets √ 1 1 d J ε = j ∈ Z : dist (εj, ∂Ω) ≥ ε d}, Ξ ≡ {ξ | − < ξj < , j = 1, . . . , d . 2 2 d

Here Ξ is the periodicity cell. Given F (ξ), a Ξ-periodic smooth function, such that F (ξ) ≥ const > 0, F (0) = −1, ∇ξ F = 0 as ξ ∈ Ξ\{0}, ξ∈∂Ξ

we define

x Qεj = {x ∈ ε (Ξ + j) | F ( ) ≤ 0} ε and consider a periodically perforated domain Ω ε = Ω\ Qεj . j∈J ε

Here and everywhere below, Ξ-periodicity means 1-periodicity with respect to ξ1 , . . . , ξd . Thus, the boundary ∂Ω ε consists of ∂Ω and the boundary of inclusions Sε ⊂ Ω, Sε = (∂Ω ε ) ∩ Ω. Let us denote by Q = {ξ | − 12 < ξj < 12 , j = 1, . . . , d, F (ξ) ≤ 0} the inclusion (hole), by S = {ξ | F (ξ) = 0} the boundary of Q, and by ν the internal unit normal to S in the extended coordinates ξ. Note that in this section we have more general micro-inhomogeneous structure. Here Q is not a ball. In what follows the summation over repeated indices is assumed. Consider the following boundary value problem: ⎧ ∂ x ∂u ε ⎪ ⎨ −Lε uε := ∂xk akj ( ε ) ∂xjε = f (x) in Ω , ∂uε (8) + p( xε )uε + εq( xε )uε = g( xε ) on Sε , ⎪ ⎩ ∂γ uε = 0 on ∂Ω,

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Gregory A. Chechkin

∂uε ε ε ε ε ε where ∂u ∂γ := akj ∂xj νk and ν = (ν1 , . . . νd ) is the unit inner normal to the boundary of the inclusions. We assume that all the functions akj (ξ), p(ξ), q(ξ) and g(ξ) are Ξ-periodic, the matrix (akj ) is symmetric, and positively defined, that is

κ1 η 2 ≤ akj ηk ηj ≤ κ2 η 2

for any vector

η,

where 0 < κ1 < κ2 < ∞. We suppose that < p(ξ)> =< g(ξ)> = 0, where < ·>S :=

S

(9)

S

· dσ.

S

For a domain G let H 1 (G) be the Sobolev space formed by all functions ◦

in L2 (G) whose gradients belong to L2 (G). Let us denote by H 1 (Ω ε , ∂Ω) the closure in H 1 (Ω ε ) of the set of smooth functions vanishing on ∂Ω. Let ◦

H0−1 (Ω ε , ∂Ω) be the dual space to H 1 (Ω ε , ∂Ω) with respect to the L2 (Ω ε ) ◦

inner product. We denote by Aε : H 1 (Ω ε , ∂Ω) → H0−1 (Ω ε , ∂Ω) the operator associated with (8). 2.2 Formal asymptotic analysis of the problem We search for the solution to (8) in the standard form [4] uε (x) ∼ u0 (x) + εu1 (x, ξ) + ε2 u2 (x, ξ) + . . . ,

ξ=

x , ε

(10)

where the functions ui (x, ξ) are assumed to be Ξ-periodic with respect to ξ. We will use the following notation (see [4]) ∂ϕ(x, ξ) ∂ ∂ϕ(x, ξ) ∂ϕ(x, ξ) akj (ξ) , := akj (ξ) νk . −Lαβ ϕ(x, ξ) := ∂αk ∂βj ∂γα ∂αj Substituting (10) in (8) and collecting the terms of order ε−1 in the equation and of order ε0 in the boundary condition on S (the highest order terms) we arrive at the following auxiliary problem: Lξξ u1 + Lξx u0 = 0 in Ξ\Q, (11) ∂u1 ∂u0 ∂γξ + ∂γx + p(ξ)u0 = g(ξ) on S. The associated integral identity takes the form ∂u1 ∂v ∂u0 ∂v akj dξ + akj dξ + p(ξ)u0 v dσ = g(ξ)v dσ, (12) ∂ξj ∂ξk ∂xj ∂ξk Ξ\Q

Ξ\Q

S

S

1 (Ξ\Q). Integral identity (12) suggests to look for the function where v ∈ Hper u1 (x, ξ) in the form


u1 (x, ξ) = L(ξ) + M (ξ)u0 (x) + Ni (ξ)

∂u0 (x) . ∂xi

195

(13)

Substituting the latter expression in (12) and collecting corresponding terms lead us to the following problems for functions Ni (ξ), M (ξ), and L(ξ): ∂Ni ∂v ∂v akj dξ + aki dξ = 0 (14) ∂ξj ∂ξk ∂ξk Ξ\Q

or, equivalently,

Ξ\Q

Lξξ (Ni (ξ) + ξi ) = 0 in Ξ\Q, ∂Ni (ξ) on S, ∂γξ = aki (ξ)νk

where i = 1, . . . , d; akj

∂M ∂v dξ + ∂ξj ∂ξk

p(ξ)v dσ = 0;

(15)

S

Ξ\Q

∂L ∂v akj dξ = ∂ξj ∂ξk

g(ξ)v dσ.

(16)

S

Ξ\Q

It is easy to verify that (9) is the solvability condition for problems (15) and (16). Note that the functions L(ξ), M (ξ), and Ni (ξ) are defined up to an additive constant that can be fixed by the following normalization condition: < L>

Ξ\Q

=< M >

Ξ\Q

=< Ni >

Ξ\Q

=0

∀ i = 1, . . . , d,

which is assumed to be fulfilled later on. Similarly, collecting the terms of order ε0 in the equation and of order ε1 in the boundary condition on S leads to the following problem for u2 (x, ξ): Lξξ u2 + Lxξ u1 + Lξx u1 + Lxx u0 = −f in Ξ\Q, (17) ∂u2 ∂u1 ∂γξ + ∂γx + p(ξ)u1 + q(ξ)u0 = 0 on S. We need the following result. Lemma 1. The functions M (ξ) and Nk (ξ) satisfy ∂M ∂u0 (x) akj dξ − pNk dσ = 0. ∂xk ∂ξj S

Ξ\Q

Proof. Substituting Ni (ξ) as a test function in (15) gives ∂M ∂Ni akj dξ + p(ξ)Ni dσ = 0. ∂ξj ∂ξk Ξ\Q

S

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Gregory A. Chechkin

Substituting M (ξ) as a test function in (14) gives ∂Ni ∂M ∂M akj dξ + aki dξ = 0. ∂ξj ∂ξk ∂ξk Ξ\Q

Ξ\Q

Since the matrix (aij ) is symmetric, we get ∂M akj dξ = pNk dσ. ∂ξj S

Ξ\Q

The lemma is proved. The weak formulation of problem (17) reads ∂u2 ∂v ∂u1 ∂v akj dξ + akj dξ + p(ξ)u1 v dσ + u0 (x) q(ξ)v dσ ∂ξj ∂ξk ∂xj ∂ξk Ξ\Q

S

Ξ\Q

−

akj

∂u0 ∂M vdξ − ∂ξj ∂xk

Ξ\Q

aij

S

∂ 2 u0 ∂Nk + aik vdξ + |Ξ\Q|f = 0 ∂ξj ∂xi ∂xk

Ξ\Q

for any v ∈ Now, in order to obtain the desired homogenized equation, we write down the solvability condition for problem (17). By applying the above lemma, this equation can be rewritten as follows: ∂ 2 u0 (x) − u0 (x) q(ξ) dσ + p(ξ)M (ξ) dσ , akj ∂xk ∂xj 1 Hper (Ω ε ).

S

S

= |Ξ\Q|f (x) +

g(ξ)M (ξ) dσ,

(18)

S

where , aik :=

aij (ξ)

∂Nk (ξ) + aik (ξ) dξ. ∂ξj

Ξ\Q

Finally, the homogenized problem reads ⎧ ∂ 2 u0 (x) ⎨ − < q>S u0 (x) + m u0 (x) = |Ξ\Q|f (x) − l , akj ∂xk ∂xj ⎩ u0 (x) = 0 on ∂Ω,

in Ω,

(19)

where m := − < pM >S , ◦

l :=< pL>S = − < gM >S .

, : H 1 (Ω, ∂Ω) → H −1 (Ω, ∂Ω) be the operator associated with the limit Let A problem (19).


197

Remark 1. The coerciveness of the limit problem (19) is a delicate question since the constant m, as we will see later, is always positive. In particular, the coerciveness of (19) is provided by the inequality m − < q>S < λ0 , aij where λ0 is the first eigenvalue of the differential operator −, ◦

Sobolev space H 1 (Ω, ∂Ω).

∂2 in the ∂xi ∂xj

2.3 Main estimates and results In this section we obtain upper and lower bounds for the coefficient m in (19) and then formulate the main statement of the paper. We begin by considering an auxiliary spectral problem of the Steklov type ⎧ ∂θ ∂ ⎪ ⎪ a = 0 in Ξ\Q, (ξ) ⎪ kj ⎪ ⎨ ∂ξk ∂ξj ∂θ (20) = Υ θ on S, ⎪ ⎪ ⎪ ∂γ ⎪ ⎩ θ(ξ) is Ξ-periodic in ξ, < θ>S = 0, where Υ is a spectral parameter. The first eigenvalue Υ1 of problem (20) can be found from the variational principle Υ1 = inf

a(ψ, ψ) < ψ 2 >S

where a(u, v) :=

akj

1 : ψ ∈ Hper (Ξ)\{0}, < ψ>S = 0 ,

∂u ∂v dξ. ∂ξj ∂ξk

Ξ\Q

The following lemma provides us with estimates of the coefficient m in the homogenized problem (19). Lemma 2. The constant m is positive. Moreover, it satisfies the estimates < p2 >S

< p2 >S a(p, p)

≤ m ≤

< p2 >S . Υ1

(21)

Remark 2. Note that the equalities in (21) are only attained if p(ξ) happens to be the first eigenfunction in (20), i.e. the eigenfunction that corresponds to the eigenvalue Υ1 . Proof. Substituting M (ξ) as a test function in (15) we have ∂M ∂M akj dξ + p(ξ)M dσ = 0. ∂ξj ∂ξk Ξ\Q

S

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Gregory A. Chechkin

Thus,

. ∂M ∂M m = − < pM >S = akj > 0, ∂ξj ∂ξk Ξ\Q

if M ≡ 0. Note that M ≡ 0 if p ≡ 0. Consider a variational problem inf 1

ψ∈Hper (Ξ)

H(ψ) ≡

!

inf 1

ψ∈Hper (Ξ)

" a(ψ, ψ) + 2 < pψ>S .

(22)

It follows from (15) that M (ξ) provides the infimum in (22). Therefore, ! " a(ψ, ψ) + 2 < pψ>S − inf 1 ψ∈Hper (Ξ)

= −a(M, M ) + 2 < pM >S = − < pM >S = m. Substituting ψ = −tp in the functional H(ψ), we get H(−tp) = t2 a(p, p) − 2t < p2 >S . To locate the minimum of H(−tp) in t, we solve the equation 0 = Ht (−t0 p) = 2t0 a(p, p) − 2 < p2 >S . This gives t0 =

< p2 >S , a(p, p)

and, hence, 2 2 2 2 2 2 S S S H(−t0 p) = −2 =− . a(p, p) a(p, p) a(p, p) This completes the proof of the first inequality in (21). Similarly, locating the minimum of H(−tϕ) in t for an arbitrary function ϕ, we get 2 < pϕ>S . H(−t0 ϕ)= − a(ϕ, ϕ) Since

m=

sup 1 (Ξ) ϕ∈Hper

2 < pϕ>S , a(ϕ, ϕ)

(23)

and the function ϕ = M brings the supremum in (23), then for an arbitrary ϕ we have 2 < pϕ>S . m≥ a(ϕ, ϕ) It follows from (23) that


1 = m

inf

1 (Ξ)\{0} ϕ∈Hper

a(ϕ, ϕ) 2 ≥ < pϕ>S

inf 1 ϕ∈Hper (Ξ)\{0},

199

Υ1 a(ϕ, ϕ) = . 2 < ϕ >S < p2 >S

< p2 >S

=0 S

Finally, m≤

< p2 >S , Υ1

where Υ1 is the first eigenvalue of the spectral problem (20). This proves the second inequality in (21). The lemma is proved. Remark 3. To give a simple explanation of the positiveness of the coefficient m arising in the homogenized problem (19), we consider the operator associated with the following modification of (8): q ≡ 0 and instead of the Dirichlet boundary condition on the exterior boundary ∂Ω the Neumann boundary condition is stated. In this case −m is the first eigenvalue of the operator associated with the respective homogenized problem. In view of the convergence of the spectrum, the first eigenvalue of the operator introduced above is close to −m for sufficiently small ε. The substitution of a constant function in the variational principle for the first eigenvalue of this operator gives zero. Thus, for any ε the said eigenvalue is negative and so is −m. √ The following theorem gives an O( ε) estimate of the discrepancy between a solution to (8) and two leading terms of expansion (10), in the Sobolev space H 1. Theorem 2. Let f (x) ∈ C 1 (Ω), and suppose that p(ξ), q(ξ), and g(ξ) are Ξ-periodic C 1 functions. Furthermore, assume that m < λ0 + < q>S ,

(24)

where λ0 is defined in Remark 1. Then for all sufficiently small ε > 0 problem (8) has a unique solution uε (x) and the following estimate holds √ (25) u0 + εu1 − uε H 1 (Ω ε ) ≤ K1 ε with a constant K1 > 0 independent of ε. Here u0 is the solution to (19) and u1 is defined by (13) with functions L(ξ), M (ξ) and Ni (ξ) constructed in (14), (15), and (16) respectively. 2.4 Auxiliary results This section is devoted to various technical results used in further analysis. The first two statements can be found in [5] and [9] and are formulated here for the sake of completeness. Their proof is omitted. Other lemmas can be found in [6].

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Gregory A. Chechkin

Lemma 3. If 1 |Ξ ∩ ω|

< q>S dξ −

Ξ∩ω

q(x, ξ) dσ ≡ 0,

(26)

S

then the following estimate holds 1 x < q>S u(x)v(x) dx − ε q(x, ) u(x)v(x) ds |Ξ ∩ ω| ε Ωε

(27)

Sε

≤ C2 ε u H 1 (Ω ε ) v H 1 (Ω ε ) . for any u(x), v(x) ∈ H (Ω ε ) with constant C2 independent of ε. 1

The next lemma allows us to neglect the right hand-side of (8) in the layer √ Πε := {x ∈ Ω : dist (x, ∂Ω) ≤ ε d} without worsening the estimate (25). Lemma 4. Let yε be the solution to ⎧ ε ε ⎨ Lε yε = −h (x) in Ω , ∂yε x x + p( ε )yε + εq( ε )yε = g( xε ) on Sε , ⎩ ∂γ yε = 0 on ∂Ω,

(28)

where hε (x) = f (x) in Πε and 0 in the remaining part of Ω ε . Then yε H 1 (Ω ε ) ≤ C3 ε.

(29)

Let λ0 be the first eigenvalue of the homogenized operator (it is defined in Remark 1). The statement below characterizes the coerciveness property of the limit problem. Lemma 5. If m < λ0 + < q>S then problem (19) is coercive. , leads Proof. The variational principle for the first eigenvalue of the operator A to the following relation ∂v ∂v , −Av, v = inf , aij + < q>S − m v 2 dx inf ◦ ◦ ∂xi ∂xj L2 (Ω) 1 1 v∈H (Ω), vL2 (Ω) =1

=

, aij

inf ◦ 1

v∈H (Ω), Ω vL2 (Ω) =1

Thus

v∈H (Ω), Ω vL2 (Ω) =1

∂v ∂v dx + < q>S − m = λ0 + < q>S − m. ∂xi ∂xj

, v −Av,

L2 (Ω)

≥ C4 v 2L2 (Ω) ,

C4 > 0,

that completes the proof. The next lemma can be considered as a modified version of Lemma 3. The functions u(ξ) and v(ξ) in its formulation are not assumed to be periodic.


201

Lemma 6. If S = 0 then p(ξ) u(ξ)v(ξ) dσ ≤ C5 ∇u L2 (Ξ) v L2 (Ξ) + u L2 (Ξ) ∇v L2 (Ξ) (30) S

for any u(ξ), v(ξ) ∈ H 1 (Ξ). Here C5 is a constant independent of ε. Proof. Since S = 0, the problem ⎧ ⎪ ⎨ ∆ξ Ψ (ξ) = 0 ⎪ ⎩ ∂Ψ = p(ξ) ∂n

in Ξ\Q, on

S,

∂Ψ =0 ∂n

(31) on ∂Ξ

has a unique solution up to an additive constant. Let us multiply the first equation in (31) by u(ξ)v(ξ) where u(ξ), v(ξ) ∈ H 1 (Ξ), and integrate it over Ξ\Q. Integration by parts on the left hand-side gives ∆ξ Ψ (ξ)u(ξ)v(ξ) dξ − p(ξ) u(ξ)v(ξ) dσ p(ξ) u(ξ)v(ξ) dσ = S

S

Ξ\Q

≤

((∇ξ Ψ (ξ)), ∇ξ (u(ξ)v(ξ))) dξ

(32)

Ξ\Q

≤ C5 ∇u L2 (Ξ) v L2 (Ξ) + u L2 (Ξ) ∇v L2 (Ξ) . The lemma is proved. The next lemma shows that the bilinear form associated with problem (8) is coercive uniformly in ε. In particular, this proves that (8) is well-posed for sufficiently small ε. Lemma 7. The coerciveness of the homogenized problem (19) implies the coerciveness of the original problem (8) for all sufficiently small ε. Proof. First, we show that x 2 1 2 p( )u (x) ds ≤ α |∇u| dx + u2 dx ε α Sε

Ωε

(33)

Ωε

for any α > 0. Indeed, by Lemma 6 we get α ε 2 2 |∇u| dξ + u2 dξ). p(ξ) u (ξ) dσ ≤ 2C5 ∇u L2 (Ξ) u L2 (Ξ) ≤ C6 ( ε α S

Ξ

Ξ

Now rewriting the latter inequality in the coordinates x = εξ and summing up over all the periodicity cells inside Ω we obtain (33).

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Gregory A. Chechkin

The next step is to verify that there exists a sufficiently large Λ independent of ε such that the operator associated with the boundary value problem ⎧ ε ⎨ Lε uε + Λuε = −f (x) in Ω , ∂uε x x + p( ε )uε + εq( ε )uε = g( xε ) on Sε , (34) ⎩ ∂γ uε = 0 on ∂Ω is coercive for any ε > 0. It follows from Lemma 3 that x ∂v ∂v x x aik ( ) dx + p( )v 2 ds + ε q( )v 2 ds + Λv 2 dx ε ∂xk ∂xi ε ε Ωε

Sε

Ωε

Sε

C7 + Λ v 2L2 (Ω ε ) . − (C7 α + + < q>S − ≥ α (35) By taking a sufficiently small α and, then, a sufficiently large Λ, we establish the positive definiteness of the bilinear form on the right hand-side of (35) and, therefore, the coerciveness. Consider the following spectral problems κ1 ∇v 2L2 (Ω ε )

O(ε)) ∇u 2L2 (Ω ε )

−1

(Aε + Λ · 1)

−1

,+Λ·1 A

ukε = µεk ukε ,

(36)

uk = µk uk ,

(37)

, is the where Aε is the operator associated with the initial problem (8) and A operator associated with the homogenized problem (19). Taking into account the coerciveness shown above, it is easy to verify that −1 −1 ,+Λ·1 satisfy conditions C1– C4 of the operators (Aε + Λ · 1) and A Theorem 1.9 from [25, Chapter III]. In particular, Theorem 1.9 implies that µε0 → µ0 as ε → 0. Let us denote by λε0 and λ0 the first eigenvalues of the , respectively. Then operators Aε and A λε0 ≡ −Λ +

1 1 → λ0 ≡ −Λ + µε0 µ0

as

ε → 0.

, implies the positive definiteness of Aε Thus, the positive definiteness of A for all sufficiently small ε. The lemma is proved. Remark 4. The analogous approach can be found in [29] (Lemma 3) and, for instance, the coercivity of Problem (34) follows from the results of this paper.


203

2.5 Justification of asymptotics Proof of Theorem 2. To estimate the H 1 -norm of the remainder u0 + εu1 − uε H 1 (Ω ε ) x x let us substitute the expression zε (x, ) = u0 (x) + εu1 (x, ) − uε (x) in (8). ε ε This gives x 1 Lε zε (x, ) = Lξx u0 (x) x + Lε u0 (x) + εLxx u1 (x, ξ) x ε ε ξ= ε ξ= ε +Lξx u1 (x, ξ)

ξ= x ε

+ Lxξ u1 (x, ξ)

ξ= x ε

1 + Lξξ u1 (x, ξ) x − Lε uε (x). ε ξ= ε

(38)

Taking into account the relations Lε uε (x) = −f (x), Lξξ u1 (x, ξ) = −Lξx u0 (x), ∂ ∂u0 (x) ∂ ∂ 2 u0 (x) −Lξx u1 (x, ξ) = aij (ξ) aij (ξ) M (ξ) + Nk (ξ) , ∂ξi ∂xj ∂ξi ∂xj ∂xk −Lxξ u1 (x, ξ) = aij (ξ)

∂u0 (x) ∂M (ξ) ∂ 2 u0 (x) ∂Nk (ξ) + aij (ξ) , ∂xi ∂ξj ∂xi ∂xk ∂ξj

(39)

and , akj

∂ 2 u0 (x) − < q>S u0 (x) + m u0 (x) = |Ξ\Q|f (x) − l ∂xk ∂xj

in Ω,

(40)

one can rewrite (38) in the domain Ω ε as follows x ∂ ∂u0 (x) aij (ξ) M (ξ) x −Lε zε (x, ) = −εLxx u1 (x, ξ) x + ε ∂ξi ∂xj ξ= ε ξ= ε ∂ ∂ 2 u0 (x) aij (ξ) + Nk (ξ) x ∂ξi ∂xj ∂xk ξ= ε ∂u0 (x) ∂M (ξ) ∂ 2 u0 (x) ∂Nk (ξ) +aij (ξ) − Lε u0 (x) (41) x + aij (ξ) ∂xi ∂ξj ξ= ε ∂xi ∂xk ∂ξj ξ= xε −

(< q>S − m) l 1 ∂ 2 u0 (x) a ˆkj u0 (x) − . + |Ξ ∩ ω| ∂xk ∂xj |Ξ ∩ ω| |Ξ ∩ ω|

Similarly, we have on Sε ∂zε (x, xε ) ∂uε (x) ∂u0 (x) ∂u1 (x, ξ) ∂u1 (x, ξ) =− + +ε x+ x ∂γ ∂γx ∂γx ∂γx ∂γξ ξ= ε ξ= ε

204

Gregory A. Chechkin

∂u0 (x) x x x ∂u1 (x, ξ) = p( )uε (x) + εq( )uε (x) − g( ) + +ε x ε ε ε ∂γx ∂γx ξ= ε ∂L(ξ) ∂M (ξ) ∂u0 (x) ∂Ni (ξ) + + u0 (x) + . ∂γξ ξ= xε ∂γξ ξ= xε ∂xi ∂γξ ξ= xε Now multiplying (41) by v(x) and integrating over Ω ε we get x − Lε zε (x, ) v(x) dx = −ε Lxx u1 (x, ξ) x v(x) dx ε ξ= ε Ωε

Ωε

∂u0 (x) ∂ aij (ξ) M (ξ) x v(x) dx ∂ξi ∂xj ξ= ε

+ Ωε

+

Ωε

∂ ∂ξi

+ + Ωε

−

1 |Ξ ∩ ω|

∂ 2 u0 (x) aij (ξ) Nk (ξ) x v(x) dx ∂xj ∂xk ξ= ε aij (ξ)

Ωε

∂u0 (x) ∂M (ξ) v(x) dx ∂xi ∂ξj ξ= xε

∂ u0 (x) ∂Nk (ξ) aij (ξ) v(x) dx − Lε u0 (x)v(x) dx ∂xi ∂xk ∂ξj ξ= xε 2

, akj

Ωε

1 ∂ 2 u0 (x) v(x) dx + ∂xk ∂xj |Ξ ∩ ω| −

l |Ξ ∩ ω|

(42)

Ωε

< q>S − m u0 (x) v(x) dx

Ωε

v(x) dx. Ωε

On the other hand, the Green formula allows to rewrite the left hand-side of (42) as follows x ∂zε v(x) ds − ∇zε ∇v(x) dx − Lε zε (x, ) v(x) dx = ε ∂γ Ωε

=

x p( )uε (x) v(x) ds + ε ε

Sε

+

+ Sε

Ωε

Sε

Sε

x q( )uε (x) v(x) ds − ε

Sε

∂u0 (x) v(x) ds + ε ∂γx

Sε

x g( )v(x) ds ε

Sε

∂u1 (x, ξ) x v(x) ds ∂γx ξ= ε

∂L(ξ) ∂M (ξ) ∂u0 (x) ∂Ni (ξ) + u0 (x) + ∂γξ ∂γξ ∂xi ∂γξ

ξ= x ε

v(x) ds

(43)


−

205

x ∇zε (x, ) ∇v(x) dx. ε

Ωε

From (42) and (43) we derive x x x ∇zε (x, ) ∇v(x) dx = p( )uε (x) v(x) ds + ε q( )uε (x) v(x) ds ε ε ε Ωε

Sε

x g( )v(x) ds + ε

− Sε

+

∂ ∂ξi

Ωε

−

Ωε

aij (ξ)

− Ωε

+

1 |Ξ ∩ ω|

Sε

∂u1 (x, ξ) x v(x) ds ∂γx ξ= ε

∂ ∂ξi

ξ= x ε

v(x) ds

(44)

∂u0 (x) aij (ξ) M (ξ) x v(x) dx ∂xj ξ= ε

∂ 2 u0 (x) aij (ξ) Nk (ξ) x v(x) dx ∂xj ∂xk ξ= ε

∂u0 (x) ∂M (ξ) v(x) dx + ε Lxx u1 (x, ξ) x v(x) dx x ∂xi ∂ξj ξ= ε ξ= ε Ωε

∂ u0 (x) ∂Nk (ξ) aij (ξ) x v(x) dx + Lε u0 (x)v(x) dx ∂xi ∂xk ∂ξj ξ= ε 2

, akj

Ωε

∂u0 (x) v(x) ds + ε ∂γx

∂M (ξ) ∂u0 (x) ∂Ni (ξ) ∂L(ξ) + u0 (x) + ∂γξ ∂γξ ∂xi ∂γξ −

Ωε

Sε

Sε

Sε

−

1 ∂ 2 u0 (x) v(x) dx − ∂xk ∂xj |Ξ ∩ ω| +

l |Ξ ∩ ω|

Ωε

< q>S − m u0 (x) v(x) dx

Ωε

v(x) dx. Ωε

In view of the obvious relation ∂ 2 u0 (x) ∂ ∂ 2 u0 (x) ∂ aij (ξ) aij (ξ) Nk (ξ) =ε Nk (ξ) ∂ξi ∂xj ∂xk ∂xi ∂xj ∂xk ξ= x ξ= x ε

−ε

∂ ∂xi

ε

∂ 2 u0 (x) aij (ξ) Nk (ξ) , ∂xj ∂xk ξ= x

ε

we have after integration by parts, ∂ 2 u0 (x) ∂ aij (ξ) Nk (ξ) x v(x) dx ∂ξi ∂xj ∂xk ξ= ε Ωε

(45)

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Gregory A. Chechkin

+ Ωε

∂ ∂ξi

∂u0 (x) ∂u1 (x, ξ) aij (ξ) M (ξ) x v(x) dx = ε x v(x) ds. ∂xj ∂γx ξ= ε ξ= ε Sε

Using (44) and the boundary condition in (11) one can obtain, after simple rearrangements, the following inequality x x x x p( ) + εq( ) zε (x, ) v(x) ds ∇zε (x, ) ∇v(x) dx + ε ε ε ε Ωε

Sε

x x x x ≤ εε q( )u1 (x, ) v(x) ds + p( )u1 (x, ) v(x) ds ε ε ε ε Sε

Sε

x 1 +ε q( )u0 (x) v(x) ds − < q>S u0 (x) v(x) dx ε |Ξ ∩ ω| Ωε

Sε

+ε Lxx u1 (x, ξ) x v(x) dx ξ=

(46)

ε

Ωε

∂u (x) ∂u (x) ∂N (x, ξ) 0 0 i + + x v(x) ds ∂γx ∂xi ∂γξ ξ= ε Sε

x 1 + p( ) u0 (x) v(x) ds + mu0 (x)v(x) dx ε |Ξ ∩ ω| Sε

+

l |Ξ ∩ ω|

v(x) dˆ x−

Ωε

Ωε

x g( ) v(x) ds ε

Sε

, akj ∂ 2 u0 (x) ∂ 2 u0 (x) ∂Nk (ξ) v(x) − aij (ξ) v(x) + |Ξ ∩ ω| ∂xk ∂xj ∂xi ∂xk ∂ξj ξ= xε Ωε

+Lε u0 (x)v(x) dx = I1 + I2 + I3 + I4 + I5 + I6 + I7 .

It remains to estimate all the terms on the right hand side. We begin with I2 . According to Lemma 3, x 1 < q>S u0 (x) v(x) dx I2 = ε q( )u0 (x) v(x) ds − ε |Ξ ∩ ω| Sε

Ωε

≤ C2 ε u0 H 1 (Ω ε ) v H 1 (Ω ε ) . By means of the same arguments as those in the proof of Lemma 3 (see [5] and [9]) it is easy to show that √ √ √ I5 ≤ C8 ε v H 1 (Ω ε ) , I6 ≤ C9 ε v H 1 (Ω ε ) , I7 ≤ C10 ε v H 1 (Ω ε ) .


207

Clearly, the terms I1 and I3 admit the following bounds |I1 | + |I3 | ≤ C11 ε v H 1 (Ω ε ) . The identity I4 ≡ 0 follows from properties of the functions Ni (ξ), i = 1, . . . , d. Substituting v = u0 + εu1 − uε in (46) and taking into account all the previous estimates we arrive at (20). The theorem is proved.

References 1. Allaire, G. Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–1518. 2. Bakhvalov, N. S. Homogenized characteristics of bodies with periodic structure. Doklady Akad. Nauk SSSR 218 (1974), 1046–1048. 3. Bakhvalov, N. S. Homogenization of partial differential equations with rapidly oscillating coefficients. Sov. Math., Dokl. 16 (1975), 351–355. 4. Bakhvalov, N. S. and Panasenko, G. P. Homogenization: Averaging Processes in Periodic Media. Mathematics and its Applications 36. Kluwer, 1989. 5. Belyaev, A. G., Pyatnitski, A. L. and Chechkin, G. A. Asymptotic Behavior of the Solution of a Boundary Value Problem in a Punctured Domain with an Oscillating Boundary. Siberian Math. J. 39 (1998), 621–644. 6. Belyaev, A. G., Pyatnitski, A. L. and Chechkin, G. A. Averaging in a Perforated Domain with an Oscillating Third Boundary Condition. Russian Acad. Sci. Sb. Mathematics 192 (2001), 933–949 7. Chechkin, G.A. and Chechkina, T.P. On Homogenization of Problems in Domains of the “Infusorium” Type. J. Math. Sci. 120 (2004), 1470–1482 8. Chechkin, G.A. and Chechkina, T.P. Homogenization Theorem for Problems in Domains of the “Infusorian” Type with Uncoordinated Structure. J. Math. Sci. 123 (2004), 4363-4380. 9. Chechkin, G. A., Friedman A. and Piatnitski A. L. The boundary value problem in domains with very rapidly oscillating boundary. J. Math. Anal. Appl. 231 (1999), 213–234. 10. Chechkin, G. A. and Piatnitski, A. L. Homogenization of boundary-value problem in a locally periodic perforated domain. Appl. Anal. 71 (1999), 215–235. 11. Chechkin, G. A., and Piatnitski, A. L., and Shamaev, A.S. Homogenization. Methods and some applications. Scientific book, Novosibirsk, 2004. (Russian) 12. Cherkaev, A. and Kohn, R. (eds.) Topics in Mathematical Modeling of Composite Materials. Birkh¨ auser, Boston, 1997. 13. Cioranescu, D. and Murat, F. Un terme etrange venu d’ailleurs. In Nonlinear Partial Differential Equations and Their Applications. Res. Notes in Math. 60 Pitman, London, 1982, 98–138. 14. Cioranescu, D. and Saint Jean Paulin, J. Truss-structures, Fourier Conditions and Eigenvalue Problems. In: Boundary Variation (J. P. Zolezio ed), Lect. Notes Control Infor. Sci. Springer-Verlag. Berlin, 1992, 6–12. 15. Cioranescu, D. and Donato, P. On a Robin Problem in Perforated Domains. In Homogenization and Applications to Material Sciences (D. Cioranescu, A. Damlamian, P. Donato eds.) GAKUTO, Gakk¯ otosho, 1997, 123–135.

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16. Cioranescu, D. and Donato, P. Homogénésation du problème de Neumann non homogène dans des ouverts perforés. Asymptotic Anal. 1 (1988), 115–138. 17. Cioranescu, D. and Saint Jean Paulin, J. Homogenization of Reticulated Structures. Springer-Verlag, Berlin, New York, 1999. 18. Gilbarg D. and Trudinger N.S. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1983. 19. Hornung, U. (ed.) Homogenization and Porous Media. IAM 6. Springer-Verlag, Berlin, 1997. 20. Jikov, V. V., Kozlov, S. M., and Oleinik, O. A. Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994. 21. Marchenko, V. A. and Khruslov, E. Ya. Boundary Value Problems in Domains with a Fine-Grained Boundary. Naukova Dumka, Kyiv, 1974. (Russian) 22. Murat, F. and Tartar, L. Calcul des variations et homog´ enéisation. Laboratoire d’Analyse Numérique, Univ. P. et M. Curie, Paris, 1984. 23. Neuss-Radu, M. Some extensions of two-scale convergence. C. R. Acad. Sci. Paris Ser.I Math. 322 (1996), 899–904. 24. Nguetseng, G. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623. 25. Oleinik, O. A., Shamaev, A. S., and Yosifian, G. A. Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam, 1992. 26. Oleinik O.A., Shaposhnikova T.A. On the Homogenization of the Poisson Equation in Partially Perforated Domain with the Arbitrary Density of Cavities and Mixed Conditions on their Boundary. Rend. Lincei: Mat. Appl., ser IX. 8 (1997), 129–146. 27. Pastukhova S.E. The averaging of a mixed problem with an oblique derivative for an elliptic operator in a perforated domain. (Russian) Diff. Uravneniya 30 (1994), 1445–1456 28. Pastukhova, S. E. On the error of averaging for the Steklov problem in a punctured domain. Diff. Uravneniya 31 (1995), 1042–1054 29. Pastukhova, S. E. Tartar’s compensated compactness method in the averaging of the spectrum of a mixed problem for an elliptic equation in a punctured domain with a third boundary condition. Mat. Sb. 186 (1995), 127–144. 30. Sanchez-Palencia, E. Homogenization Techniques for Composite Media. Springer-Verlag, Berlin, 1987. 31. Sukretny˘ı, V. I. Asymptotic expansion of the solutions of the third boundary value problem for second-order elliptic equations in punctured domains. Mat. Fiz. Neline˘ın. Mekh. 7, (1987), 67–72 32. Sukretny˘ı, V. I. The problem of steady heat conduction in perforated regions. Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 84-48, 1984. 33. Volkov, D. B. Averaging of some boundary value problems in domains with a periodic structure. U.S.S.R. Comput. Math. Math. Phys. 22 (1, 1982), 117–128

Homogenization of random non stationary parabolic operators Andrey Piatnitski Narvik University College Postboks 385, 8505 Narvik, Norway and P.N. Lebedev Physical Institute of RAS, Leninski pr., 53, Moscow 119991, Russia e-mail: [email protected]

Introduction We study homogenization problem for a random non stationary parabolic second order equation of the form x t 1 x t ∂ ε ε u (x, t) = div a( , 2 )∇u (x, t) + g( , 2 )uε (x, t) + f (x, t), (1) ∂t ε ε ε ε ε with a small positive parameter ε. This model equation describes various processes in a medium with spatial microstructure whose characteristics are rapidly changing functions of time. Throughout this article we assume that the spatial microstructure is periodic and that the characteristics of this microstructure are random stationary rapidly oscillating processes. The presence in the equation of a large zero order term, linear or nonlinear, leads to rather unusual asymptotic behaviour of a solution of (1), as ε tends to zero. We will show that almost sure (a.s.) homogenization result in general fails to hold and that a weaker averaging result takes place. Namely, under certain mixing conditions, a solution of (1) converges in law in a suitable functional space to a solution of a homogenized stochastic partial differential equation (SPDE). Our aim is to justify this convergence and to investigate the properties of the limit SPDE. The presence of a large factor in the lower order terms of the equation is natural when studying long term behaviour of solutions. We illustrate this with the following example. Many applications deal with parabolic operator of the form ∂ v(y, s) = div (a(y, s)∇v(y, s)) + εg(y, s)v(y, s), ∂t

210

Andrey Piatnitski

with a small potential εg(y, s), here ε characterizes the range of oscillation of the potential. In order to study the behaviour of solutions at large time s ∼ ε−2 , one can make the diffusive change of variables x = εy, t = ε2 s. In the new coordinates the equation reads x t x t ∂ v = div a( , 2 )∇v + ε−1 g( , 2 )v, ∂t ε ε ε ε it is similar to the equation (1). First rigorous homogenization results for random elliptic and parabolic operators in divergence form were obtained in the works [5], [10]. After that this topic has been studied by many mathematicians, now it is well presented in the existing literature. However, some important problems in the field remain open. It is known that, in contrast with periodic case, the presence of lower order terms in the equation with random coefficients might change crucially the effective behaviour of solutions. In this work we consider an intermediate case of equations with lower order terms whose coefficients are periodic in spatial variables and random in time. Averaging problems for these equations with diffusive driving process were studied in [2] in linear case, and in [12] in nonlinear case. The convection diffusion problem of this type with a generic stationary driving process having good mixing properties, have been considered in [6].

1. The setup This section is devoted to homogenization of equations of the form x t 1 x t ∂ ε ε u (x, t) = div a( , 2 )∇u (x, t) + g( , 2 )uε (x, t) + f (x, t), ∂t ε ε ε ε ε

(2)

with generic random stationary in time and periodic in spatial variables coefficients. For this equation we consider a Cauchy problem in Rn × (0, T ) with the initial condition (3) uε (x, 0) = u0 (x). Here and later on we assume that u0 ∈ L2 (Rn ) and f ∈ L2 ((0, T ) × Rn ). Remark 1. The Cauchy problem has been chosen for the sake of definiteness. Initial boundary problems with Dirichlet or Neumann conditions can be studied in a similar way. Problem (2)–(3) will be investigated under the following assumptions on the coefficients. H1.The coefficients aij (y, s) and g(y, s) are [0, 1]n periodic in y.

Homogenization of random non stationary parabolic operators

211

H2. The functions aij (y, s) and g(y, s) are stationary random functions of s defined on a probability space (Ω, F, P), with values in the space of periodic functions of y. We assume that aij (y, s) = aij (y, s, ω) and g(y, s) = g(y, s, ω) are measurable with respect to the σ-algebra B(Tn ) × B(−∞, +∞) × F, where the symbol B stands for the Borel σ-algebra. For simplicity we assume that Ω is equipped with a random dynamical system Tt and that ˇij (y, Ts ω), aij (y, s, ω) = a

g(y, s, ω) = gˇ(y, Ts ω),

(4)

where a ˇij (y, ω) and gˇ(y, ω) are given random function with values in L∞ (Tn ). Let us recall that Tt is a group of measurable transformations Tt : Ω −→ Ω such that - Ts1 Ts2 = Ts1 +s2 , T0 = Id; - Ts preserves measure P for any s ∈ R, i.e. P(Ts (G)) = P(G) for any G ∈ F; - Ts (ω) is a measurable map from (Ω ×R, F ×B) to (Ω, F), where B stands for a Borel σ-algebra. H3.Uniform ellipticity: aij (y, τ )ζi ζj ≥ λ|ζ|2 , |aij (y, τ )| ≤ λ−1 ,

λ > 0,

|g(y, s)| ≤ λ−1

for all y, τ and ζ ∈ Rn . H4.Centering condition. The average of g(z, s) is equal to zero that is g(z, s)dz = 0 E

(5)

[0,1]n

for all s ∈ R. In order to formulate one more assumption we first recall the definition of mixing coefficients. Let ξs be a stationary random process defined on a probability space (Ω, F, P), and denote F≤t = σ{ξs , s ≤ t} and F≥t = σ{ξs , s ≥ t}. The function κ(γ) defined by κ(γ) =

sup

E1 ∈F≤t , E2 ∈F≥(t+γ)

|P(E1 )P(E2 ) − P(E1 ∩ E2 )|,

is called strong mixing coefficient of the process ξ· . Notice that since ξ· is stationary, κ(γ) does not depend on t. The function ϕ(γ) defined by P(E1 ∩ E2 ) ϕ(γ) = sup P(E1 ) − P(E2 ) , E1 ∈ F≤t , E2 ∈ F≥(t+γ) P(E2 ) = 0

212

Andrey Piatnitski

is called the uniform mixing coefficient of ξ· . The function ρ(γ) defined by E((η1 − Eη1 )(η2 − Eη2 )) , ρ(γ) = sup η1 , η2 Eη12 Eη22 η1 ∈ L2 (Ω, F≤t , P), η2 ∈ L2 (Ω, F≥(t+γ) , P). is called the maximum correlation coefficient of ξ· . We now consider σ-algebras F≤t and F≥t , generated by the coefficients a(y, t), g(y, t) of operator (2), and impose the following condition on the corresponding mixing coefficients H5.At least one of the following conditions holds true. ∞

∞ κ(γ)dγ < ∞,

0

∞ ϕ(γ)dγ < ∞,

0

ρ(γ)dγ < ∞ 0

Remark 2. Condition H4 can be assumed without loss of generality. Indeed, the relation (5) can be achieved by means of the following factorization of unknown function in (2) u ˜ε (x, t) = exp(< g¯ > /ε)u(x, t),

with < g¯ >= E

g(z, s)dz.

[0,1]n

If < g¯ >= 0, then the homogenization takes place on the background of exponential growth or decay of the solution. Under conditions H1-H3 problem (2)-(3) is well posed for each ε > 0. Lemma 1. Let H1-H3 be fulfilled. Then for each ε > 0 problem (2)-(3) has a unique solution uε ∈ L2 (0, T ; H 1 (Rn )) ∩ C(0, T ; L2 (Rn )) for all ω ∈ Ω. This solution defines a measurable mapping uε : (Ω, F) −→ (L2 (0, T ; H 1 (Rn )) ∩ C(0, T ; L2 (Rn )), B). The estimate holds uε C((0,T );L2 (Rn )) + uε L2 ((0,T );H 1 (Rn )) ≤ C(ε)( f L2 (0,T ;H −1 (Rn ) + u0 L2 (Rn ) ).

(6)

Proof. The existence and the uniqueness of a solution as well as a priori estimate (6) are standard. The measurability is the consequence of the fact that uε depends continuously on the data of the problem.


213

2. Factorization of the equation It is convenient to represent g(y, s) as a sum g(y, t) =< g > (t) + g˜(y, t),

def

< g > (t) =

g(z, t)dz, [0,1]n

and to introduce a new unknown function v ε = v ε (x, t) as follows t s 1 ε ε u (x, t) = v (x, t) exp < g > 2 ds , ε 0 ε It is straightforward to check that v ε satisfies the equation x t 1 x t ∂ ε ε v (x, t) = div a( , 2 )∇v (x, t) + g˜( , 2 )v ε (x, t) ∂t ε ε ε ε ε s 1 t < g > 2 ds , +f (x, t) exp − ε 0 ε

(7)

(8)

v ε (x, 0) = u0 (x). This problem will be studied in the following sections. In the remaining part of this section we deal with the exponential factor in (7). Lemma 2. Suppose that at least one of the conditions H5 holds. Then the process t s 1 ε (9) < g > 2 ds ζt = ε ε 0

satisfies functional Central Limit Theorem (invariance principle) with zero mean and the diffusion given by ∞ 2

σ =2

E < g > (0) < g > (s)ds.

(10)

0

That is the process {ζtε } converges in law in the space C[0, ∞) to the process {σWt }, where Wt is a standard Brownian motion. The proof of this statement can be found for instance in [9], Chapter 9. As a consequence of the lemma we obtain the convergence t s 1 L exp < g > 2 ds −→ exp(σWt ) (11) ε 0 ε in the space C[0, ∞).

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3. A priori estimates for the factorized equation In this section we derive a priori estimates for a solution of problem (8) and of more general Cauchy problem of the form x t 1 x t ∂ ε ε z (x, t) = div a( , 2 )∇z (x, t) + g˜( , 2 )z ε (x, t) + h(x, t), (12) ∂t ε ε ε ε ε z ε (x, 0) = z0 (x), which involves a nontrivial right hand side. Proposition 1. A solution z ε of problem (12) admits an estimate z ε L∞ (0,T ;L2 (Rn )) + z ε L2 (0,T ;H 1 (Rn )) ≤ C( z0 L2 (Rn ) + h L2 (0,T ;H −1 (Rn ) ) (13) with a constant C which does not depend on ε. Proof. By construction the function g˜(y, s) has zero average in variable y for ˜ = g˜ is solvable in the space of periodic all s and ω. Therefore, the equation ∆Q ˜ q (y, s)| ≤ C. g L∞ < ∞, we have |˜ functions. Denote q˜ = ∇y Q. Since ˜ Clearly, q(y, s) satisfies the relation divy q˜(y, s) = g˜(y, s). In coordinates x = εy, t = ε2 s it reads εdivx q˜ε (x, t) = g˜ε (x, t);

(14)

Here and afterwards for a generic function F (y, s) we use the notation x t Fε (x, t) = F ( , 2 ), ε ε

∂ ∂ t Fε (x, t) = F (y, 2 ) x , ∂yi ∂yi ε y= ε

(15)

x ∂ ∂ Fε (x, t) = F ( , s) t ∂s ∂s ε s= 2 ε Multiplying the equation (12) by z ε and integrating the resulting relation over the set Rn × (0, T ) gives (z ε (x, t))2 dx − (z0 (x))2 dx Rn Rn t aij,ε (x, τ )∇z ε (x, τ ) · ∇z ε (x, τ )dxdτ + =− n 0 R t t −1 g˜ε (x, τ )(z ε (x, τ ))2 dxdτ + z ε (x, τ )h(x, τ )dxdτ. +ε 0

Rn

0

Rn

Considering (14), after multiple integration by parts we get


t (z ε (x, t))2 dx + aij,ε (x, τ )∇z ε (x, τ ) · ∇z ε (x, τ )dxdτ 0 Rn Rn t (z0 (x))2 dx + z ε (x, τ )˜ qε (x, τ ) · ∇z ε (x, τ )dxdτ + = n n 0 R R t ε z (x, τ )h(x, τ )dxdτ. +

215

0

(16)

Rn

Denote the right hand side here by Rε (t). For each γ > 0 we have t (z0 (x))2 dx + γ −1 |z ε (x, τ )˜ qε (x, τ )|2 dxdτ |Rε (t)| ≤ Rn

t +γ ≤

Rn

+γ

Rn

0

0

−1

0

t

|∇z ε (x, τ )|2 dxdτ +

(z0 (x))2 dx + Cγ −1

Rn

t

|z ε (x, τ )|2 dxdτ + γ

t

h(·, τ ) 2H −1 (Rn ) dτ

z ε (·, τ ) H 1 (Rn ) h(·, τ ) H −1 (Rn ) dτ 0

t 0

Rn

0

t

+γ 0

Rn

Rn

|∇z ε (x, τ )|2 dxdτ

(|z ε (·, τ )|2 + |∇z ε (·, τ )|2 )dxdτ.

It remains to combine this bound with (16). The desired estimate (13) now follows from Gronwall lemma.

4. Auxiliary problems Passage to the limit in problem (8) requires introducing a number of auxiliary functions usually called correctors. These correctors will be defined as solutions of auxiliary parabolic equations. This section is devoted to those auxiliary problems and their properties. ∂ aij (y, s) ∂y∂ j and consider in a cylinder Tn × (−∞, +∞) the Denote A = ∂y i following two equations: ∂ ∂ j χ (y, s) − Aχj (y, s) = aij (y, s) ∂s ∂yi

(17)

∂ G(y, s) − AG(y, s) = g˜(y, s) ∂s

(18)

Proposition 2. Equations (17) and (18) have stationary solutions in the space L∞ (−∞, +∞; C(Tn ))∩L2loc (−∞, +∞; H 1 (Tn )). Each of these solutions is unique up to a (random) additive constant. The estimates χ L2 (N,N +1;H 1 (Tn )) ≤ C,

G L2 (N,N +1;H 1 (Tn )) ≤ C,

(19)

hold uniformly in N ∈ R. Moreover, the constant C is deterministic and only depends on λ in H3.

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Proof. Let us show that (17) has a stationary solution. To this end we consider the following Cauchy problem ∂ ∂ j χ (y, s) − AχjN (y, s) = aij (y, s)1[N,N +1) (s), ∂s N ∂yi

(y, s) ∈ Tn × (N, +∞)

χjN (y, N ) = 0. (20) For s < N we set χjN (y, s) = 0. By the Nash estimates (see [8]), the function χjN is continuous and satisfies the upper bound χjN L∞ ((−∞,N +1]×Tn ) ≤ c1 (γ). Since the right hand side in (20) is equal to zero for all s > N + 1, by the maximum principle the last estimate is valid for all s: χjN L∞ ((−∞,+∞)×Tn ) ≤ c1 (γ). We want to show that χjN (y, s) decays exponentially as (s − N ) → ∞. Lemma 3. There are nonrandom independent of N constants c2 (λ) > 0 and c3 (λ) > 0 such that |χjN (y, s)| ≤ c2 exp(−c3 (s − N ))

(21)

Proof of the lemma. Notice that Tn

χjN (y, s)dy = 0

(22)

for each s ∈ R. Indeed, integrating the equation (17) on the cylinder (s1 , s2 ) × Tn , one has Tn χjN (y, s1 )dy = Tn χjN (y, s2 )dy. Then (22) follows from the equality χjN (y, N ) = 0. By the Poincaré inequality, considering (22) and H3, we get for each s ∂ k ∂ k aij (y, s) χN (y, s) χN (y, s)dy ≥ (23) ∂y ∂y n i j T −1 k 2 −1 |∇y χN (y, s)| dy ≥ c4 λ |χkN (y, s)|2 dy. ≥λ Tn

Tn

χjN (y, s)

Now we multiply (17) by and integrate the resulting relation on the cylinder (Tn × (s1 , s2 ). This gives s2 χkN (·, s) 2L2 (Tn ) ds (24) χkN (·, s2 ) 2L2 (Tn ) − χkN (·, s1 ) 2L2 (Tn ) ≤ −c4 λ−1 s1

for all s1 and s2 such that N + 1 ≤ s1 ≤ s2 . This implies the bound


217

χkN (·, s) 2L2 (Tn ) ≤ c2 exp(−c3 (s − (N + 1)) It is also clear that the function χkN (·, s) 2L2 (Tn ) is monotone on the interval (N + 1, ∞). To complete the proof of the lemma it remains to apply once again the Nash inequality. We define a vector function χ(y, s) by χk (y, s) =

+∞

χkN (y, s)

(25)

N =−∞

By construction and in view of the last Lemma, χj (y, s) solves the equation (17) and satisfies the estimate (19). We want to show that χj (y, s) is stationary. The fact that any finite dimensional distribution of this random function is invariant with respect to all integer shifts easily follows from the stationarity of aij (·, s). Taking in the above procedure an arbitrary rational step size q instead of 1, we construct a solution of equation (17) whose finite dimensional distributions are invariant with respect to any shift of the form kq with integer k. It is easy to check that this new solution coincides with χj (y, s). Thus, by arbitrariness of q, the finite dimensional distributions of χj (y, s) are invariant with respect to any rational shift. Now the stationarity of χj (y, s) follows by continuity arguments. The uniqueness of a stationary solution up to a (random) additive constant follows from Lemma 3. Indeed, if we assume the existence of two distinct stationary solutions with zero average, then their difference vanishes as s → ∞. This contradicts the stationarity. We impose the following normalization conditions for χk (y, s) and G(y, s): j χ (y, s)dy = 0, G(y, s)dy = 0 (26) Tn

Tn

This makes the choice of the corresponding additive constants unique. We set ˜ ω) = G(y, 0, ω). χ ˜k (y, ω) = χk (y, 0, ω), G(y, Since χk (y, s) and G(y, s) are continuous in s, the functions χ ˜k and G are well defined. By definition aij (y, s + τ, ω) = aij (y, s, Tτ ω). Therefore, considering the uniqueness of solution of problem (17), we have χk (y, s + τ, ω) = χk (y, s, Tτ ω). In particular, ˜k (y, Ts ω). χk (y, s, ω) = χk (y, 0, Ts ω) = χ Similarly, ˜ Ts ω). G(y, s, ω) = G(y, 0, Ts ω) = G(y,

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5. Homogenization of the factorized equation We begin by considering the equation (2) in the particular case f = 0. Our aim is to show that in this case factorized problem (8) admits a.s. homogenization. Theorem 1. Let f = 0. Then under our standing assumptions a solution v ε of problem (8) converges a.s., as ε → 0, in the space L∞ (0, T ; L2 (Rn )) towards a solution of the following Cauchy problem ∂ 0 ˆ 0 (x, t), v (x, t) = div a ˆ∇v 0 (x, t) + ˆb · ∇v 0 (x, t) + Gv ∂t v 0 (x, 0) = u0 (x). The homogenized equation has constant coefficients defined by ∂ j χ (y, s) dy, a îj = E aik (y, s) δkj + ∂yk Tn ∂ ˆbi = E g˜(y, s)χi (y, s) + aij G(y, s) dy, ∂yj Tn ˆ = E g˜(y, s)G(y, s)dy. G

(27)

(28) (29) (30)

Tn

Proof. Assume for a while that u0 ∈ C0∞ . Then a solution v 0 of problem (27) is a C ∞ function which vanishes at infinity, as well as its partial derivatives, faster than any negative power of (1 + |x|). We then substitute the following ansatz x t x t vˇε (x, t) = v 0 (x, t) + εχ( , 2 ) · ∇v 0 + εG( , 2 )v 0 ε ε ε ε in the equation (8). Considering (17) and (18), after straightforward rearrangements we get x t ∂ ε ε ε ε (ˇ v (x, t) − v (x, t)) − div a( , 2 )∇(ˇ v (x, t) − v (x, t)) ∂t ε ε 1 x t v ε (x, t) − v ε (x, t)) − g˜( , 2 )(ˇ ε ε ε x t ∂ 1 ∂ k x t ∂ 0 1 ∂ x t ∂ ∂ 0 χ ( , 2) G( , 2 )v 0 + εχk ( , 2 ) = v0 + v + v ∂t ε ∂s ε ε ∂xk ε ∂s ε ε ε ε ∂t ∂xk x t ∂ x t ∂2 1 ∂ x t ∂ 0 1 x t 0 +εG( , 2 ) v 0 −aij ( , 2 ) v0 − aij ( , 2 ) v − g˜( , 2 )v ε ε ∂t ε ε ∂xi ∂xj ε ∂yi ε ε ∂xj ε ε ε −

1 ∂ x t ∂ k x t ∂ 0 x t ∂ k x t ∂2 aij ( , 2 ) χ ( , 2) v − aij ( , 2 ) χ ( , 2) v0 ε ∂yi ε ε ∂yj ε ε ∂xk ε ε ∂yi ε ε ∂xk ∂xj


−

∂ ∂yi

x t x t aij ( , 2 )χk ( , 2 ) ε ε ε ε

219

∂2 x t x t ∂3 v 0 − εaij ( , 2 )χk ( , 2 ) v0 ∂xk ∂xj ε ε ε ε ∂xk ∂xi ∂xj

x t x t ∂ 0 1 ∂ x t ∂ x t −˜ g ( , 2 )χk ( , 2 ) v − aij ( , 2 ) G( , 2 )v 0 ε ε ε ε ∂xk ε ∂yi ε ε ∂yj ε ε x t ∂ x t ∂ 0 ∂ x t x t ∂ 0 aij ( , 2 )G( , 2 ) −aij ( , 2 ) G( , 2 ) v − v ε ε ∂yi ε ε ∂xj ∂yi ε ε ε ε ∂xj x t x t ∂2 x t x t −εaij ( , 2 )G( , 2 ) v 0 − g˜( , 2 )G( , 2 )v 0 ε ε ε ε ∂xi ∂xj ε ε ε ε ∂ 0 x t ∂2 x t ∂ k x t ∂2 v − aij ( , 2 ) v 0 − aij ( , 2 ) χ ( , 2) v0 ∂t ε ε ∂xi ∂xj ε ε ∂yi ε ε ∂xk ∂xj ∂ x t x t ∂2 x t x t ∂ 0 aij ( , 2 )χk ( , 2 ) − v 0 − g˜( , 2 )χk ( , 2 ) v ∂yi ε ε ε ε ∂xk ∂xj ε ε ε ε ∂xk x t ∂ x t ∂ 0 ∂ x t x t ∂ 0 aij ( , 2 )G( , 2 ) −aij ( , 2 ) G( , 2 ) v − v ε ε ∂yi ε ε ∂xj ∂yi ε ε ε ε ∂xj =

x t x t x t ∂ ∂ 0 x t ∂ −˜ g ( , 2 )G( , 2 )v 0 + εχk ( , 2 ) v + εG( , 2 ) v 0 ε ε ε ε ε ε ∂t ∂xk ε ε ∂t x t x t ∂3 x t x t ∂2 −εaij ( , 2 )χk ( , 2 ) v 0 − εaij ( , 2 )G( , 2 ) v0 . ε ε ε ε ∂xk ∂xi ∂xj ε ε ε ε ∂xi ∂xj ∂ 0 Substituting the right hand side of the equation (27) in place of ∂t v gives x t ∂ ε (ˇ v (x, t) − v ε (x, t)) − div a( , 2 )∇(ˇ v ε (x, t) − v ε (x, t)) ∂t ε ε

1 x t − g˜( , 2 )(ˇ v ε (x, t) − v ε (x, t)) ε ε ε x t x t ∂ j x t ∂ x t i x t akj ( , 2 )χ ( , 2 ) = a îj − aij ( , 2 ) − aik ( , 2 ) χ ( , 2 )− ε ε ε ε ∂yk ε ε ∂yk ε ε ε ε 2 x t x t x t ∂ x t ∂ v 0 + ˆbi − g˜( , 2 )χj ( , 2 ) − aji ( , 2 ) G( , 2 ) × ∂xi ∂xj ε ε ε ε ε ε ∂yi ε ε ∂ x t x t ∂ 0 ˆ − g˜( x , t )G( x , t ) v 0 − aij ( , 2 )G( , 2 ) v + G ∂yi ε ε ε ε ∂xj ε ε2 ε ε2 x t ∂ ∂ 0 x t ∂ +εχk ( , 2 ) v + εG( , 2 ) v 0 ε ε ∂t ∂xk ε ε ∂t x t x t ∂3 x t x t ∂2 −εaij ( , 2 )χk ( , 2 ) v 0 − εaij ( , 2 )G( , 2 ) v0 . ε ε ε ε ∂xk ∂xi ∂xj ε ε ε ε ∂xi ∂xj Denote the right hand side of the last formula by R2ε . Since the expressions in the figure brackets are periodic in spatial variables and have zero average for any t, they can be represented as in (14):

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a îj − aij (y, s) − aik (y, s)

∂ j χ (y, s) ∂yk

∂ akj (y, s)χi (y, s) = divy κ1,ij (y, s) − ∂yk ˆbi − g˜(y, s)χj (y, s) − aji (y, s) ∂ G(y, s) ∂yi ∂ − (aij (y, s)G(y, s)) = divy κ2,i (y, s) ∂yi ˆ − g˜(y, s)G(y, s) = divy κ3 (y, s) G where the functions κ1 , ij(y, s), κ2 , i(y, s) and κ3 (y, s) are periodic in y and satisfy the estimates κ1,ij L2 ((s,s+1)×Tn ) ≤ C, κ2,i L2 ((s,s+1)×Tn ) ≤ C, κ3 L2 ((s,s+1)×Tn ) ≤ C, uniformly in s ∈ R. Then R2ε takes the form x t ∂2 x t ∂ 0 x t R2ε = εdivx κ1,ij ( , 2 ) v 0 + εdivx κ2,i ( , 2 ) v + εdivx κ3 ( , 2 )v 0 ε ε ∂xi ∂xj ε ε ∂xi ε ε x t ∂ ∂ 0 x t ∂ +εχk ( , 2 ) v + εG( , 2 ) v 0 (31) ε ε ∂t ∂xk ε ε ∂t x t x t ∂3 x t x t ∂2 v 0 − εaij ( , 2 )G( , 2 ) v0 . −εaij ( , 2 )χk ( , 2 ) ε ε ε ε ∂xk ∂xi ∂xj ε ε ε ε ∂xi ∂xj Due to the properties of v 0 this implies the estimate R2ε L2 ((0,T );H −1 (Rn )) ≤ Cε. Therefore, by Proposition 1 ˇ v ε − v ε L2 ((0,T );H 1 (Rn )) ≤ Cε,

ˇ v ε − v ε L∞ ((0,T );L2 (Rn )) ≤ Cε.

(32)

Combining the latter estimate with an evident bound ˇ v ε − v 0 L∞ (Rn ×(0,T )) ≤ Cε we obtain the desired statement for all smooth u0 with compact support. In order to prove this result for general u0 ∈ L2 (Rn ) we introduce a family of functions uδ0 ∈ C0∞ (Rn ) such that uδ0 − u0 L2 (Rn ) ≤ δ. If we denote v δ,ε a solution of problem (8) with initial condition uδ0 , then according to Proposition 1 the estimate v δ,ε − v ε L∞ (0,T ;L2 (Rn )) ≤ Cδ holds. Evidently, we have v δ,0 −v 0 L∞ (0,T ;L2 (Rn )) ≤ Cδ. As was proved above, v δ,ε converges, as ε → 0, to v δ,0 in L∞ (0, T ; L2 (Rn )). Therefore, lim sup v ε − v 0 L∞ (0,T ;L2 (Rn )) ≤ Cδ. ε→0

Since δ is an arbitrary positive number, the result follows.


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Remark 3. In the proof of Theorem 1 we did not use the mixing conditions H5, but only the ergodicity of the dynamical system Ts . The statement of this theorem remains valid for any ergodic dynamical system Ts without mixing assumptions. In particular, we obtain the following result. Corollary 1. Let the coefficients of problem (2)-(3) be periodic in spatial variables and stationary ergodic in time, and suppose the uniform ellipticity cong˜(y, s)dy = 0 for all s ∈ R a.s. ditions H3. Assume, furthermore, that Tn

Then problem (2)-(3) admits a.s. homogenization and the limit operator is a non random parabolic operator with constant coefficients given by (28)-(30). It should be noted that the methods developed in the proof of Theorem 1 apply to the equation (12) with a right hand side h(x, t) ∈ L2 ((0, T ) × Rn ). The following result holds true. Theorem 2. Let h(x, t) ∈ L2 ((0, T ) × Rn ) and z0 ∈ L2 (Rn ). Then a solution of problem (12) converges a.s., as ε → 0, in the space L2 (0, T ; H 1 (Rn )) towards a solution of problem ∂ 0 ˆ 0 (x, t) + h(x, t), z (x, t) = div a ˆ∇z 0 (x, t) + ˆb · ∇z 0 (x, t) + Gz ∂t z 0 (x, 0) = z0 (x), (33) with constant non random coefficients given by (28)–(30). We proceed by studying problem (8) with non trivial right hand side. We denote s 1 t ε ζt = < g > 2 ds. ε 0 ε and introduce V ε (x, t) to be a solution to the following Cauchy problem in Rn × (0, T ) ∂ ε ˆ ε (x, t) + f (x, t) exp(−ζtε ), V (x, t) = div (ˆ a∇V ε (x, t)) + ˆb · ∇V ε (x, t) + GV ∂t (34) V ε (x, 0) = u0 (x), with coefficients defined in (28)–(30). It is convenient to represent a solution v ε of problem (8) as a sum v ε = V ε + (v ε − V ε ). We will show that (v ε − V ε ) tends to zero in probability, as ε → 0, in the norm of L2 ((0, T ) × Rn ), while V ε converges in law in L2 ((0, T ) × Rn ) to a solution of Cauchy problem ∂ 0 ˆ 0 (x, t) + f (x, t) exp(−σWt ), v (x, t) = div a ˆ∇v 0 (x, t) + ˆb · ∇v 0 (x, t) + Gv ∂t (35) v 0 (x, 0) = u0 (x), with σ given by (10). Notice that this equation has a random right hand side.

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Proposition 3. The L2 ((0, T ) × Rn ) norm of the difference (v ε − V ε ) tends to zero in probability as ε → 0. Proof. By Lemma 2 the process ζtε converges in law in C(0, T ), as ε → 0, towards σWt . Therefore, a random function f (x, t)ζtε converges in law in L2 ((0, T ) × Rn ) to σf (x, t)Wt . By the Prokhorov theorem this implies that for any δ > 0 there is a compact set K δ ⊂ L2 ((0, T ) × Rn ) such that P{f (x, t)ζtε ∈ K δ } ≤ δ. Consider a finite δ-net in K δ , for which we use the notation {hj }, j = 1, 2, . . . , N (δ), and denote zjε (x, t) and zj0 (x, t) respectively solutions of problem (12) and (33) with right hand side hj (x, t) and initial condition u0 (x). By Theorem 1 for any δ > 0 there exists ε0 (δ) > 0 such that max P{ zjε − zj0 L2 > δ}
(2C + 1)δ} N (δ) (δ) N ! " Ej + P Ej ∩ V ε − v ε L2 > (2C + 1)δ ≤P Ω\ j=1

j=1

N (δ)

δ+

j=1

N (δ)

P{ zj0 − zjε L2 > δ} ≤ δ +

j=1

δ = 2δ. N (δ)

This implies the required convergence in probability. Proposition 4. The function V ε converges in law, as ε → 0, in L2 ((0, T ) × Rn ) towards a solution v 0 of problem (35). Proof. Notice that a solution of problem (33) as a functional of the right hand side defines a continuous mapping from L2 ((0, T )×Rn ) to L2 ((0, T ); H 1 (Rn ))∩ L∞ (0, T ; L2 (Rn )). Then the convergence in law of the right hand side in (33) implies the convergence in law, in the corresponding functional space, of V ε , and the desired statement follows.


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We summarize the above assertions in the following theorem. Theorem 3. Under conditions H1–H5 the solution of factorized problem (8) converges in law, as ε → 0, in the strong topology of the space L2 ((0, T ) × L2 (Rn )) towards a solution of problem (35) whose coefficients are given by (28)–(30). Proof. This statement is a consequence of Propositions 3 and 4.

6. Homogenization of the original equation We now turn to the homogenization of the original problem (2)–(3). Notice first that the random process (exp(ζtε ), exp(−ζtε )) converges in law in (C[0, T ])2 to the process (exp(σWt ), exp(−σWt )), where ζtε and σ are defined in (9) and(10) respectively. The asymptotic behaviour of a solution to problem (2)–(3) is described by the following Theorem 4. Let conditions H1–H5 be fulfilled. Then, as ε → 0, a solution uε of problem (2)–(3) converges in law in the strong topology of L2 (Rn × (0, T )) to a solution of the following stochastic partial differential equation ∂2u ˆ ∂u ˆ + ˆbi + gû ˆ dt + σ u ˆ dWt + f (x, t), dˆ u= a îj ∂xi ∂xj ∂xi

(36)

u ˆ(x, 0) = u0 (x), ˆ given by (28)–(30); σ is defined in (10). ˆ + 1 σ 2 and a îj , ˆbi and G with gˆ = G 2 According to [3] problem (36) is well posed and has a unique solution. Hence, the limit law is well defined. Remark 4. If Tn g(z, s)dz = 0 for almost all s then σ is equal to zero and the limit problem (36)is deterministic. As was already mentioned, in this case uε converges a.s. Proof (Theorem 4). The solution uε of problem (2)–(3) can be written as a sum uε (x, t) = (v ε (x, t) − V ε (x, t)) exp(ζtε ) + V ε (x, t) exp(ζtε ), where v ε and V ε satisfies (8) and (34) respectively, and ζtε is defined in (9). By Proposition 3, the factor (v ε (x, t) − V ε (x, t)) in the first term on the right

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hand side converges in probability to zero in L2 (Rn × (0, T )) norm. Since by Lemma 2 the function ζtε converges in law in the space C[0, T ], the product (v ε (x, t) − V ε (x, t))ε(ζtε ) tends to zero in probability in L2 (Rn × (0, T )). The function V ε as an element of L2 (Rn ×(0, T )), depends continuously on the trajectories of the process ζ·ε in the topology of C(0, T ), so does the product V ε (x, t) exp(ζtε ). Therefore, convergence in law of the process ζ·ε towards σW· implies convergence in law of the expression V ε exp(ζ·ε ) to the function v 0 exp(σW· ), where v 0 is a solution to problem (35). It remain to show that v 0 exp(σW· solves the homogenized equation (36). To this end we denote u ˆ = v 0 exp(σW· ) and Aˆ = a îj

∂2 ∂ ˆ + ˆbi + G, ∂xi ∂xj ∂xi

and consider for arbitrary ϕ(x) ∈ C0∞ (Rn ) the inner product (ˆ u(t), ϕ) taken in L2 (Rn ). This expression defines a diffusion process in R. Applying Ito’s formula to this process gives d(ˆ u, ϕ) = exp(σWt )d(v 0 (t), ϕ) + σ(v 0 (t), ϕ) exp(σWt )dWt 1 ˆ 0 (t), ϕ)dt + σ 2 (v 0 (t), ϕ) exp(σWt )dt = exp(σWt )(Av 2 1 u(t), ϕ)dWt + σ 2 (ˆ u(t), ϕ)dt (f (x, t), ϕ) exp(σWt ) exp(−σWt )dt + σ(ˆ 2 û(t), ϕ)dt + 1 σ 2 (ˆ = (Aˆ u(t), ϕ)dt + (f (x, t), ϕ)dt + σ(ˆ u(t), ϕ)dWt . 2 ˆ is a solution Considering also an evident relation u ˆ(0) = u0 we conclude that u of problem (36). According to [3] this problem has a unique solution, thus the limit law is uniquely defined. In the end of this section we formulate similar results for initial boundary problems. Given a Lipschitz domain Q ⊂ Rn , consider in the cylinder Q × (0, T ) a Dirichlet initial boundary problem of the form x t 1 x t ∂ ε ε u (x, t) = div a( , 2 )∇u (x, t) + g( , 2 )uε (x, t) + f (x, t), (37) ∂t ε ε ε ε ε uε (x, t) = 0 on ∂Q × (0, T ),

u(x, 0) = u0 (x),

where u0 ∈ L (Q) and f ∈ L (Q × (0, T )). Under assumption H3 for each ε > 0 the existence and the uniqueness of a solution of this problem in the space L2 ((0, T ); H01 (Q)) ∩ C((0, T ); L2 (Q)) are well known, see, for instance, [8]. The statement below can be justified in the same way as that for the case of Cauchy problem. We omit its proof. 2

2


225

Theorem 5. Let conditions H1–H5 be fulfilled. Then, as ε → 0, a solution uε of problem (37) converges in law in the strong topology of L2 (Q × (0, T )) to a solution of the limit (homogenized) stochastic partial differential equation which has the form ∂2u ˆ ∂u ˆ + ˆbi + gû ˆ dt + σ u ˆ dWt + f (x, t), (38) dˆ u= a îj ∂xi ∂xj ∂xi u ˆ(x, t) = 0 on ∂Q × (0, T ),

u ˆ(x, 0) = u0 (x).

All the coefficients of this equation are the same as in Theorem 4. Similar results hold true for Neumann and Fourier initial boundary problems.

7. Equations with diffusion driving process In this section we consider an important particular case of a diffusion finite dimensional driving process in (1). Then problem (2)–(3) reads x 1 x ∂ ε u (x, t) = div a( , ξ t2 )∇uε (x, t) + g( , ξ t2 )uε (x, t) + f (x, t), (39) ∂t ε ε ε ε ε uε (x, 0) = u0 (x); here and afterwards ξs is a stationary diffusion process with values in Rd . We denote the generator of this process by L: L = qkm (y)

∂2 + B(y) · ∇y . ∂yk ∂ym

The advantages of operators with diffusion driving processes are - the coefficients of homogenized problem can be found in terms of solutions of non random elliptic auxiliary problems; - sufficient conditions for mixing properties required in H5 can be formulated explicitly in terms of the coefficients of generator L. We suppose the following conditions to hold A1. The coefficients a(z, y), g(z, y) and q(y) are uniformly bounded as well as their derivatives: there exists C > 0 such that for all (z, y) ∈ Tn × Rd |aij (z, y)| + |∇z aij (z, y)| + |∇y aij (z, y)| ≤ C , |g(z, y)| + |∇z g(z, y)| + |∇y g(z, y)| ≤ C , |qkm (y)| + |∇y qkm (y)| ≤ C , for all (z, y) ∈ Tn × Rd and for all 1 ≤ i, j ≤ n, 1 ≤ k, l ≤ d; the symbols ∇z and ∇y stand for the gradients with respect to z and y respectively. The vector function B and its derivatives satisfy polynomial growth condition:

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|B(y)| + |∇B(y)| ≤ C(1 + |y|)µ for some µ ≥ 0 and C > 0. A2. Matrices aij and qkm are uniformly positive definite: there is λ > 0 such that λ |z |2 ≤ aij (z, y)zi zj , ∀z ∈ Rn , λ |y |2 ≤ qkm (z, y)yk ym , ∀y ∈ Rd .

A3. There exist constants α > −1, R > 0 and C > 0 such that b(y) · y ≤ −C|y|α for all y ∈ {y : |y| ≥ R}. |y| n A4. Centering condition: E T g(z, ξs )dz = 0. As was proved in [11], under conditions A1–A3 the process ξs has a unique invariant measure in Rd whose density solves the problem ρ(y)dy = 1; L∗ ρ = 0, Rd

the notation L∗ is used for the adjoint operator. Moreover, for any N > 0 there is CN > 0 such that ρ(y) ≤ CN (1 + |y|)−N . It was also shown in the same work that the strong mixing coefficient of a stationary version of the diffusion process ξs possesses the property H4. Denote L2ρ (Tn × Rd ) the weighted L2 space with the norm 2 f (z, y) ρ = f 2 (z, y)ρ(y)dydz, Tn R d

and Hρ1 (Tn × Rd ) = {f ∈ L2ρ (Tn × Rd ) : |∇z f | + |∇y f | ∈ L2ρ (Tn × Rd )}. Also we will use the notation A = divz a(z, y)∇z . The proof of following two technical statements can be found in [2]. Lemma 4. Let f ∈ L2ρ (Tn × Rd ), and suppose that |f (z, y)| ≤ C (1 + |y|)p ∀(z, y) ∈ Tn × Rd for some C > 0 and p ∈ R and that f (z, y)ρ(y)dydz = 0. Then the equation

Tn R d


227

(A + L) u(z, y) = f (z, y) has a solution in the space of functions of polynomial growth in y: |u(z, y)| ≤ C (1 + |y|)p+1 ∀(z, y) ∈ Tn × Rd This solution is unique up to an additive constant. If, in addition, there is N > 0 such that for all n1 , n2 ∈ N with n1 + n2 ≤ N we have n n ∂z 1 ∂y 2 f (z, y) ≤ C (1 + |y|)p ∀(z, y) ∈ Tn × Rd , then

n n ∂z 1 ∂y 2 u(z, y) ≤ C (1 + |y|p+1 ) ∀(z, y) ∈ Tn × Rd .

Proposition 5. For any t > 0, µ > 0, γ > 0 and β > 0 the relation holds lim E sup εβ |ξs/εγ |µ = 0. ε→0

s≤t

We proceed with averaging procedure. As above we represent g(z, ξs ) as the sum g(z, ξs ) =< g > (ξs ) + g˜(z, ξs ) with < g > (ξs ) = Tn g(z, ξs )dz. In order to construct the limit operator we need two correctors which are defined as solutions of the following auxiliary equations (A + L)χj (z, y) = −

∂ aij (z, y), ∂zi

(40)

(A + L)G(z, y) = −˜ g (z, y). By Lemma 4 these equations have solutions in growth in y. The main result of this section is

Hρ1 (Tn

(41) × R ) of polynomial d

Theorem 6. Under assumptions A1–A4 the solution uε of problem (39) converges in law, as ε → 0, in the space L2 ((0, T ) × Rn ) to a solution u0 of the following stochastic PDE a∇u0 (x, t)) − ˆb∇u0 (x, t)ˆ g u0 (x, t) dt + σu0 (x, t)dWt , du0 (x, t) = div(ˆ u0 (x, 0) = u0 ,

where

a(z, y)(Id + ∇z χ(z, y))ρ(y)dzdy,

a ˆ= ˆb =

Tn

Rd

(g(z, y)χ(z, y) + a(z, y)∇z G(z, y))ρ(y)dzdy, Tn R d

228

Andrey Piatnitski

gˆ = σ2 = Rd

Tn R d

1 g(z, y)G(z, y)ρ(y)dzdy + σ 2 , 2

(42)

2q(y) ∇y G(z, y))dz · ∇y G(z, y))dz ρ(y)dy, Tn

Tn

and Wt is a standard 1D Wiener process. Proof. Since all the conditions of Theorem 4 are fulfilled, we need not prove the convergence of solutions of problem (39) to a solution of a limit stochastic PDE but only the fact that the effective coefficients given by (42) coincide with those defined in Theorem 4. First we show that the corresponding diffusion coefficients σ are identical. The validity of the Central Limit Theorem for stationary processes of the form F (ξs ), F ∈ Lq (Rd ), with diffusion ξs satisfying condition A3, have been justified in [11]. In particular, it has been proved that for the process < g(ξs ) > the corresponding variance σ is given by the formula ¯ ¯ · ∇G(y)ρ(y)dy, σ 2 = 2q(y)∇G(y) Rd

¯ is a solution to the equation LG(y) ¯ where G =< g > (y). Since operator L applied to a function F (z, y) commutes with taking the average of F in z, we obtain ∇ < g > (y) = ∇ Tn G(z, y)dz, and the desired coincidence follows. We proceed with the other coefficients. For any ϕ ∈ C ∞ (0, T ; C0∞ (Rn )) consider the auxiliary process H ε (t) = (ϕ, v ε ) + ε(χε · ∇ϕ, v ε ) + ε(Gε , ϕ); here and afterwards for a generic function F = F (z, y) we use the notation F ε (x, t) = F ( xε , ξt/ε2 ); (·, ·) stands for the inner product in L2 (Rn ). Applying Ito’s formula to H ε (t) gives dH ε = {(aεij ∂xi ∂xj ϕ, v ε ) + ε−1 (∂zi aεij ∂xi ϕ, v ε )}dt

√ g ε ϕ, v ε )dt + (∇y Gε ϕ + ∇y χεi ∂xi ϕ, v ε ) · q ε dBt +ε−1 (˜ + ε−1 (Lχε ∇ϕ, v ε ) + (g ε χε ∇ϕ, v ε ) + (∂t ϕ, v ε ) +ε−1 (Aχε ∇ϕ, v ε ) + ε(χε ∂t ∇ϕ, v ε ) +(∂zi (aεij χk )ε ∂xj ∂xk ϕ + aεij ∂zi χεk ∂xj ∂xk ϕ, v ε ) +ε(aε χε ∇∇∇ϕ, v ε ) + ε−1 (LGε ϕ, v ε ) + (Gε g ε ϕ, v ε ) +ε−1 (AGε ϕ, v ε ) + (∂zi (aεij Gε )∂xj ϕ, v ε ) + (aεij ∂zi Gε ∂xj ϕ, v ε ) +ε(aε Gε ∇∇ϕ, v ε ) dt.

Taking into account the equations (40), (41) we get


229

dH ε = {(aεij ∂xi ∂xj ϕ, v ε ) + (∂zi (aεij χk )ε ∂xj ∂xk ϕ + aεij ∂zi χεk ∂xj ∂xk ϕ, v ε ) +(g ε χε ∇ϕ, v ε ) + (Gε g ε ϕ, v ε ) + (∂t ϕ, v ε )

+(∂zi (aεij Gε )∂xj ϕ, v ε ) + (aεij ∂zi Gε ∂xj ϕ, v ε ) dt √ (∇y Gε ϕ + ∇y χεi ∂xi ϕ, v ε ) · q ε dBt + εdRε (t),

(43)

where Bt is a standard n-dimensional Brownian motion, and Rε satisfies the estimate E sup |Rε (t)| ≤ C. t≤T

This estimate follows from Proposition 1. For the stochastic term on the right hand side of (43) we have Lemma 5. The bound holds t ∇y Gε (·, s)ϕ(·, s) lim E sup ε→0 t≤T 0

+∇y χi ε(·, s)∂xi ϕ(·, s), v ε (·, s) · q ε (s)dBs = 0.

Proof. Since

Tn

∇y χ(z, y)dz = 0 and

Tn

∇y G(z, y)dz = 0, we have

t ∇y Gε (·, s)ϕ(·, s) + ∇y χεi (·, s)∂xi ϕ(·, s), v ε (·, s) · q ε (s)dBs E sup t≤T 0

t Ji1,ε (·, s), ∂xi (ϕ(·, s)v ε (·, s)) = εE sup t≤T 0

ε · q (s)dBs

2,ε (·, s), ∂xj (∂xi ϕ(·, s)v ε (·, s)) +Jij

where J 1 (z, y) and J 2 (z, y) are periodic in z functions such that divz J 1 (z, y) = ∇y G(z, y),

divz J 2 (z, y) = ∇y χ(z, y).

The statement of the lemma now follows from (1) and the Burkholder-DavisGundy inequality. To complete the proof of the theorem we notice that by virtue of (1) and the Birkhoff theorem t ε lim E sup(ϕ(·, t), v (·, t)) − (ϕ(·, 0), u0 ) − (∂s ϕ(·, s), v ε (·, s))ds ε→0

t≤T

0

t ¯ a îj ∂xi ∂xj ϕ(·, s) + ˆb∇ϕ(·, s) + Gϕ(·, s) , v ε (·, s) ds = 0, − 0

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Andrey Piatnitski

where the coefficients a ˆ, ˆb have been defined in Theorem 6 and ˆ= G g(z, y)G(z, y)ρ(y)dzdy. Tn R d

References 1. Bensoussan A., Lions J.-L., Papanicolaou G. Asymptotic Analysis of Periodic Structures. North-Holland Publ. Comp. 1978. 2. Campillo F., Kleptsyna M. L., Piatnitski A. L. Homogenization of random parabolic operator with large potential, Stoc. Processes and Appl., 93, 2001, p. 57-85. 3. Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. 4. Jikov V. V., Kozlov S. M., Oleinik O. A. Homogenization of Differential Operator and Integral Functionals. Springer-Verlag, 1994. 5. Kozlov, S. M. The averaging of random operators. Math. USSR Sbornik, 109, 1979, p. 188-202. 6. Kleptsyna M. L., Piatnitski A. L. Homogenization of a random non-stationary convection-diffusion problem. Russian Mathematical Surveys, 57, 2003, p. 95118. 7. Kleptsyna M. L., Piatnitski A. L. Homogenization of random parabolic operators. in Homogenization and applications to material sciences. Cioranescu D. et al. (eds.), Gakkotosho. GAKUTO Int. Ser., Math. Sci. Appl., 9, p. 241-255, 1997. 8. Ladyzhenskaia O. A., Solonnikov V. V., Ural’tseva N. N. Linear and quasilinear equations of parabolic type. Nauka, Moscow, 1967. 9. Liptser R. Sh., Shiryaev A. N. Theory of Martingales. Mathematics and its Applications. Kluwer, Dordrecht, 1989. 10. Papanicolaou, G., Varadhan, S. R. S. Boundary values problems with rapidly oscillating random coefficients. Random fields. Rigorous results in statistical mechanics and quantum field theory. Esztergom, 1979, Coloq. Math. Soc. Janos Bolyai 27, 1981, p. 835-873. 11. Pardoux, E., Veretennikov, A. Yu. On the Poisson equation and diffusion approximation, I. Ann. Probab., 29, 2001, p. 1061-1085. 12. Pardoux E., Piatnitski A. L. Homogenization of a nonlinear random parabolic partial differential equation. Stoc. Processes and Appl., 104, 2003, p. 1-27.

Part II

Problems with concentration

Γ -convergence for concentration problems Adriana Garroni Dipartimento di Matematica, Universit` a di Roma “La Sapienza” Piazzale A. Moro 2, 00185 Roma, Italy e-mail: [email protected], http://www.mat.uniroma1.it/people/garroni

1 Introduction In these notes we will consider a large class of variational problems in which a concentration phenomenon occurs. This is a typical phenomenon in problems with scaling invariance as, for instance, semilinear problems involving the critical Sobolev exponent. More precisely we will study the asymptotic behaviour of problems of the form ∗ F (u) dx : |∇u|2 dx ≤ ε2 , u = 0 on ∂Ω , (1) SεF (Ω) := sup ε2 Ω

Ω

when ε → 0+ , where Ω is a bounded domain of Rn , n ≥ 3, and F is a ∗ nonnegative upper-semicontinuous function bounded from above by c|t|2 , 2n being the critical Sobolev exponent. with 2∗ = n−2 If F is a smooth function the maximizers of (1) satisfy the following semilinear equation in Ω, −∆u = λε f (u) u=0 on ∂Ω, where f = F and the Lagrange multiplier λε tends to +∞ when ε → 0+ . The, by now, classical approach for this type of semilinear problems is the concentration-compactness alternative due to P.L. Lions [21]. For the case of ∗ smooth F with critical growth (e.g. F (t) = |t|2 ) he proved that the sequence of maximizers either concentrates at a single point (in a sense that will be clear in Section 2) or is compact in H01 (Ω). In particular in the case Ω = Rn one can exclude compactness and deduce that only concentration is allowed. As a particular case of problem (1) with, possibly, non smooth (or degenerate) F , we can also obtain interesting free boundary problems as the plasma problem or the Bernoulli free boundary problem (see for instance [16], [14] and [9]). The latter can be consider as the weak Euler-Lagrange equation of the following variational problem

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Adriana Garroni

∗ SεV = sup ε−2 |A| : A ⊆ Ω , cap(A, Ω) ≤ ε2 , where cap(A, Ω) denotes the harmonic capacity of the set A. The critical ∗ case, corresponding to the choice F (t) = |t|2 , and the Bernoulli problem will be described in details in Section 2 and can be considered as two “extreme” particular cases for problem (1). In order to deal with the general problem (1), the concentration-compactness alternative has been generalized by Flucher and M¨ uller in [13]. Again one deduce concentration for the sequence of maximizers (or almost maximizers). In this notes we propose a different method to deduce concentration and, more in general, to study the asymptotic behaviour of problem (1). We will use the notion of Γ -convergence which is the natural convergence for functionals in order to deduce convergence of extrema. The idea of Γ -convergence is to substitute a sequence of functionals {Fε } by an effective Γ -limit functional F which captures the relevant features of the sequence {Fε }. In particular, sequences of (almost) maximizers converge to maximum points of F. Therefore, in terms of Γ -convergence, we study the asymptotic behaviour of the functional −2∗ F (εu) dx , Fε (u) = ε Ω

with the constraint Ω |∇u|2 dx ≤ 1 and u ∈ H01 (Ω). Analyzing the Γ -limit F we describe the asymptotic behaviour of Fε (uε ) along all weakly converging sequences and we also deduce concentration. The approach of Γ -convergence for this kind of concentration phenomena is recent and has been successfully used also in the study of Ginzburg-Landau problem by Alberti, Baldo and Orlandi [1]. A delicate point, in general for Γ convergence and here in particular, is the choice of the topology which should be strong enough in order to assure convergence of maxima and weak enough in order to detect concentration. A second natural question for this type of concentration results is whether it is possible to characterize the concentration point, in particular whether its position can be influenced by the shape of the domain. A crucial role in the identification of the concentration point is played by the Green’s function for the Dirichlet problem with the Laplacian. More precisely we will see that the maximizing sequences concentrates at the harmonic centers of Ω (the minima of the Robin function; i.e., the diagonal of the regular part of the Green’s function).

2 Two classical examples We introduce the problem starting with two examples: the Sobolev inequality and the harmonic capacity.

Γ -convergence for concentration problems

235

2.1 Sobolev inequality Let us fix a domain Ω ⊂ Rn , with n ≥ 3. We know that the Sobolev space 2n ; i.e., there H01 (Ω) is embedded continuously in Lp (Ω) for every p ≤ 2∗ = n−2 exists a constant Cp (Ω), depending on p and Ω, such that

|u| dx ≤ Cp (Ω)

|∇u| dx

p

Ω

2

p2 (2)

Ω

for every u ∈ H01 (Ω). The embedding is also compact for p < 2∗ . It is well known that for p = 2∗ (the so-called critical case) the embedding is not compact. There is a large number of interesting and difficult analytical and geometrical problems involving the critical growth and many interesting phenomena arise from this lack of compactness. Let us consider first the Sobolev inequality in Rn

2∗

|u| dx ≤ S

∗

Rn

|∇u| dx 2

22∗

∀u ∈ C0∞ (Rn ) .

,

Rn

(3)

Inequality (3) holds true for all functions in the Deny space D1,2 (Rn ) obtained as the closure of C0∞ (Rn ) with respect to the topology induced by ∇uL2

the L2 norm of the gradient (D1,2 (Rn ) = C0∞ (Rn ) Sobolev constant.

) and S ∗ is the best

Question 1 Is S ∗ achieved? In other words, does there exist a function u ∈ D1,2 (Rn ) such that (3) holds with equality? In the case of Rn the answer is yes. This problem has been solved by Talenti in ’76 ([23]), and all the possible solutions have been completely characterized. Namely, all the solutions of the following variational problem ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ⎪ ⎪ |u|2 dx ⎨ ⎬ n ∗ 1,2 n R : u ∈ D (R ) S = max , ∗ 22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ |∇u| ⎩ ⎭ Rn

or equivalently S ∗ = max

∗

|u|2 dx : u ∈ D1,2 (Rn ) and Rn

|∇u|2 ≤ 1

,

(4)

Rn

have been characterized. Indeed, one can consider the Euler-Lagrange equation of (4) and, by a rearrangement argument, prove that

236

Adriana Garroni

u1 (x) =

1 (c2

+ |x|2 )

n−2 2

,

(5)

is a solution. Here the constant c is a renormalization which gives |∇u1 |2 dx = 1 . Rn

All other solutions can be obtained by scaling and translating u1 . Indeed, ∗ both the Dirichlet integral on Rn and the L2 norm are invariant under the following dilations and translations: n x−y for σ > 0 and y ∈ Rn . u(x) = σ − 2∗ u1 σ 2 2∗ ∗ | and |u(x)|2 = σ −n |u1 x−y | , and Namely |∇u(x)|2 = σ −n |∇u1 x−y σ σ thus, by a change of variables, ∗ ∗ |∇u|2 dx = |∇u1 |2 dx and |u|2 dx = |u1 |2 dx . Rn

Rn

Rn

Rn

Let us consider now the case Ω = Rn , for instance consider the case Ω bounded. In this case H01 (Ω) = D1,2 (Ω) and the Sobolev inequality reads

2∗

∗

|u| dx ≤ S (Ω) Ω

|∇u| dx 2

22∗

∀u ∈ H01 (Ω) .

(6)

Ω

Remark 1. Again by a scaling argument one can see that the best Sobolev constant does not depend on the domain; i.e., S ∗ (Ω) = S ∗

∀ Ω ⊆ Rn .

Question 2 Is S ∗ achieved in Ω ? The answer in this case is no. In fact, otherwise, we would find solutions for (4) with compact support and hence not of the form (5). Thus an other question arise naturally. Question 3 What we can expect from an optimal sequence; i.e., a sequence uε ∈ H01 (Ω) such that ∗ |uε |2 dx ∗ Ω (7) 2∗ = S + o(1) ? |∇uε |2 dx Ω

2


237

We may have a rather precise idea of the situation through the following n example: take the sequence vε such that vε (|x|) = ε− 2∗ u1 ( xε ) and Ω = BR (BR denotes the ball centered in the origin of radius R). Clearly we have ∗ ∗ |vε (|x|)|2 dx = lim |u1 |2 dx = S ∗ . lim ε→0

ε→0

BR

BεR ∗

In particular we get that the sequence |vε |2 converges to S ∗ δ0 (δ0 denotes the Dirac mass at zero) weakly∗ in the sense of measure. It is then easy to see that uε (x) := (vε (|x|) − vε (R))+ ∈ H01 (Ω) satisfies (7). Moreover also uε ∗ ∗ ∗ concentrates all the energy at zero; i.e., |uε |2 S ∗ δ0 and |∇uε |2 δ0 . The scaling invariance of the problem is responsible for this phenomenon of concentration and in general for the lack of compactness in the embedding theorem. This lack of compactness can be very well described, and this has been done by P.L. Lions ([21]) by means of the concentrationcompactness principle. Generally speaking, it consists in the analysis of the possible ways a bounded sequence of measures can loose compactness. In the special case of the Sobolev embedding theorem he proves in particular the following Concentration-compactness alternative. We fix a sequence uε ∈ D1,2 (Ω), with ∇uε L2 ≤ 1. Up to a subsequence there exists a function u0 ∈ D1,2 (Ω) such that uε u0

∗

in L2 (Ω)

∇uε ∇u0

and

in L2 (Ω) ;

i.e., uε u0 in D1,2 (Ω). We may also assume that there exist two measures µ∗ , ν ∗ ∈ M(Ω) := (C(Ω)) , such that ∗

∗

|uε |2 dx ν ∗

in M(Ω)

and

∗

|∇uε |2 dx µ∗

in M(Ω) .1

In order to study the possible lack of compactness for uε the idea of P.L. Lions is to characterize the measures ν ∗ in terms of µ∗ . ∗

∗

Remark 2. Note that if ν ∗ = |u0 |2 dx then we have compactness in L2 for uε , while if we also know that µ∗ = |∇u0 |2 dx we conclude strong convergence of uε in D1,2 (Ω). In general, by the lower semi-continuity of the norm, we get µ∗ ≥ |∇u0 |2 dx . Thus we can isolate the atoms of µ∗ , {xi }i∈J , and rewrite µ∗ as follows 1

∗

Recall: we say that a sequence of measures µε µ in M(Ω) if and only if

lim

ε→0

ϕ dµε = Ω

ϕ dµ Ω

∀ ϕ ∈ C(Ω) .

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Adriana Garroni

µ∗ = |∇u0 |2 dx +

µi δxi + µ %,

(8)

i∈J

% is the non-atomic where µi denotes the positive weight of the atom xi and µ part of µ∗ − |∇u0 |2 dx. Remark 3. Note that µ % in general may also contain a part which is absolutely continuous with respect to the Lebesgue measure. Lemma 1 ([21]) Let uε , u0 ∈ D1,2 (Ω) be such that Ω |∇uε |2 ≤ 1, uε u0 ∗ ∗ ∗ in D1,2 (Ω), |uε |2 dx ν ∗ and |∇uε |2 dx µ∗ in M(Ω) for some measures µ∗ and ν ∗ . Assume µ∗ be decomposed as in (8); then ∗ 1. there exist non-negative constants νi such that ∗ i) ν ∗ = |u0 |2 dx + νi∗ δxi ; ∗

∗

ii) |uε − u0 |2 dx

i∈J

νi∗ δxi in M(Ω);

i∈J n

iii) 0 ≤ νi∗ ≤ S ∗ (µi ) n−2 for all i ∈ J. 2. (Alternative) If ν ∗ (Ω) = S ∗ and µ∗ (Ω) = 1; i.e., uε is an optimal sequence for the Sobolev embedding, then one of the two following situations is possible a) Concentration: there exists x0 ∈ Ω such that µ∗ = δx0 and ν ∗ = S ∗ δx0 ; ∗ b) Compactness: ν ∗ = |u0 |2 dx. Note that in the alternative b), by the optimality of uε we also get µ∗ = |∇u0 |2 dx and hence uε → u0 in D1,2 (Ω). Remark 4. 1. In the case Ω = Rn , by the fact that the Sobolev constant is never achieved in Ω, we deduce that only alternative a) is possible; i.e., concentration occurs. 2. Part 1 of Lemma 1 is obtained by a fine use of the Sobolev inequality. In particular note that ii) states that the only way a bounded sequence ∗ in D1,2 (Ω) can loose compactness in L2 (Ω) is by concentration (so an oscillating bounded sequence for which the gradients weak converge but ∗ do not concentrate, is always strongly convergent in L2 (Ω)). 2.2 Capacity and isoperimetric inequality for the capacity We now introduce the notion of harmonic capacity for which we will see that a similar concentration phenomenon arises.


239

Definition 2 (Capacity) Given Ω ⊆ Rn and an open set A ⊂ Ω, the capacity of A with respect to Ω is the following set function |∇u|2 dx : u ≥ 1 a.e. in A , u ∈ H01 (Ω) . (9) cap(A, Ω) = inf Ω

Note that the constraint u ≥ 1 a.e. in A in the definition of the capacity is convex and strongly closed in H01 (Ω) and hence it is closed in the weak topology. So that there exists a unique minimum point for problem (9). It is called the capacitary potential of A with respect to Ω. Remark 5. 1. By a truncation argument we can show that the capacitary potential uA satisfies uA = 1 a.e. in A. 2. The capacitary potential is also a weak solution of the following EulerLagrange equation ⎧ ⎪ in Ω \ A ⎨−∆u = 0 (10) u=1 on ∂A ⎪ ⎩ u=0 on ∂Ω . The capacity theory is a classical tool in potential theory. In the variational approach it is the natural object when studying problems in the Sobolev space H01 (Ω). It is involved in regularity results for elliptic problems, fine behaviour of Sobolev functions, etc. (for a very nice review of this subject see Frehse [17]). In general it is not very easy to compute explicitly the capacity of a set. Let us consider the easiest case: two concentric balls. Example 1. Fix 0 < r < R and compute the capacity of Br with respect to BR . From now on we will denote by K(ρ) =

γn ρn−2

with γn =

1 (n − 2)|S n−1 |

the fundamental singularity of the Laplacian ∆ in Rn , n ≥ 3, and then K(|x − y|) will be the fundamental solution with singularity at y. In particular K(|x|) is harmonic outside zero. Moreover K(|x|) − K(R) = 0 on ∂BR and K(|x|)−K(R) K(r)−K(R) = 1 on ∂Br . Thus we have that the capacitary potential of Br with respect to BR is given by K(|x|) − K(R) ,1 . u(x) = min K(r) − K(R) Then, using the fact that −∆K(|x|) = δ0 in the sense of distributions, we get

240

Adriana Garroni

1 cap(Br , BR ) = |∇u| dx = K(r) − K(R) BR 1 . = K(r) − K(R)

2

BR \Br

∇K∇u dx

Similarly, we obtain cap(Br , Rn ) =

1 . K(r)

The notion of capacity can be extended to any subset E of Ω as follows cap(E, Ω) = inf{cap(A, Ω) : A open , A ⊇ E} . Proposition 3 Let A and B be two given subsets of Ω. The following properties hold. 1. (Monotonicity) If A ⊂ B then cap(A, Ω) ≤ cap(B, Ω) ; 2. (Subadditivity) cap(A ∪ B, Ω) ≤ cap(A, Ω) + cap(B, Ω) ; 3. (Scaling property) For any ρ > 0 we have cap(ρA, ρΩ) = ρn−2 cap(A, Ω) ; 4. If B is open and A ⊆ B ⊆ Ω, then 1 1 1 ≥ + cap(A, Ω) cap(A, B) cap(B, Ω) and equality holds if and only if B is a superlevel of the capacitary potential uA of A in Ω. The above properties can be easily deduced from the definition of the capacity and making use of the capacitary potentials. Note that in particular the capacity is an external measure, but in general it is not additive (it can be proved that it is a measure only on the class of zero capacity sets). The “equivalent” of the Sobolev inequality in the case of the capacity is the so-called isoperimetric inequality for the capacity: there exists a constant S V (Ω) such that |A| ≤ S V (Ω) (cap(A, Ω))

2∗ 2

∀ A⊆Ω.

(11)

Question 4 Is this constant S V (Ω) achieved by a non-trivial set A? In other words: there exists a subset A of Ω with cap(A, Ω) > 0, such that equality holds in (11)?


241

If Ω = Rn the answer is again yes. In fact we can also compute it explicitly. It is given by S V := S V (Rn ) = max {|A| : cap(A, Rn ) ≤ 1} . It easy to see, by a symmetrization argument, that the maximum is achieved on balls; i.e. A = BR . Imposing the constraint cap(BR , Rn ) = 1 the radius R can be computed explicitly and hence the constant S V turns out to be the following n

S V = ωn γnn−2 ,

(12)

where γn = K(1) and ωn = |B1 |. Remark 6. Note that, thanks to the scaling property of the capacity (see Proposition 3, 4)), the equality in the isoperimetric inequality (11) is achieved for all balls. Indeed, if R is such that cap(BR , Rn ) = 1 and |BR | = S V , then n

|BρR | = ρn |BR | = S V ρn = S V (cap(ρBR , Rn )) n−2 . In the case Ω = Rn the answer is no. The constant S V is not achieved by a non trivial set A for a reason which is similar to the one we saw in the Sobolev embedding case. Indeed also in this case it can be seen, using the scaling property of the capacity, that ∀ Ω.

S V (Ω) = S V

Moreover whenever cap(Rn \ Ω, Rn ) = 0 and cap(A, Ω) = 0, it can be seen, by using the maximum principle, that cap(A, Ω) > cap(A, Rn ) and then, for any such A, the equality in (11) is not possible. Thus also in this case we may wonder which is the behaviour of an optimal sequence; i.e., a sequence of sets Aε such that |Aε | n

(cap(Aε , Ω)) n−2

= S V + o(1) .

(13)

So, as above, to have an idea let us construct an optimal sequence by choosing Aε = Brε , with rε → 0. Then, for any R > 0, (cap(Aε , Ω))

n n−2

= (cap(Brε , Ω))

n n−2

=

r n ε

R

n n−2 R cap(BR , Ω) . rε

Now it can be proved that lim cap(BR ,

ε→0

R Ω) = cap(BR , Rn ) , rε

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Adriana Garroni

thus, if we choose R such that |BR | = S V , we clearly have that the sequence satisfies (13). Notice that the balls Brε shrink to the origin. Again we have a concentration phenomenon for the capacitary potentials similar to the case of the Sobolev embedding. More in general a way to construct an optimal sequence for (11) is to consider the following variational problem " ! ∗ (14) SεV (Ω) = ε−2 max |A| : cap(A, Ω) = ε2 , ∗

where the factor ε−2 is the right scaling passing from the capacity to the volume.

3 The general problem We are now in a position to introduce a very general class of problems which includes and unifies the two examples shown above as two extreme case of the same phenomenon. 3.1 Variational formulation We will consider the following family of variational problem depending on a small parameter ε > 0 ∗ F (u) dx : u ∈ D1,2 (Ω) , |∇u|2 dx ≤ ε2 , (15) SεF (Ω) = ε−2 sup Ω

Ω

with the following assumptions: ∗

i) 0 ≤ F (t) ≤ c|t|2 for every t ∈ R; ii) F ≡ 0 and upper semi-continuous. To simplify the exposition we also assume iii) there exist the following two limits F0 (t) = lim

t→0

F (t) |t|2∗

and F∞ (t) = lim

t→∞

F (t) . |t|2∗

With this formulation we recover the two examples seen before.


243

Examples. ∗

1. If F (t) = |t|2 , then (15) gives the Sobolev embedding problem and SεF (Ω) = S ∗ for every Ω ⊆ Rn . 2. We are allowed to choose F discontinuous, then we recover the case of the capacity taking F of the form 0 if t < 1 F (t) = 1 if t ≥ 1. We claim that, with this choice of F , problem (15) coincides with (14); i.e., V 1,2 (16) |∇u|2 dx ≤ ε2 Sε (Ω) = sup |{u ≥ 1}| : u ∈ D (Ω) , Ω

Clearly SεV (Ω) ≥ sup |{u ≥ 1}| : u ∈ D1,2 (Ω) , |∇u|2 dx ≤ ε2

;

Ω

indeed any u ∈ D1,2 (Ω) satisfying Ω |∇u|2 dx ≤ ε2 is a good competitor for the computation of the capacity of the set A = {u ≥ 1} and gives cap(A, Ω) ≤ Ω |∇u|2 dx ≤ ε2 . On the other hand, if A is an open set such that cap(A, Ω) ≤ ε2 and uA is the capacitary potential of A, then A ⊆ {uA ≥ 1} and Ω |∇uA |2 dx = cap(A, Ω) ≤ ε2 ; hence we get (16). 3. If F ∈ C 1 we can compute the Euler-Lagrange equation and we have −∆u = λε f (u) in Ω u=0 on ∂Ω, where f = F and λε is the Lagrange multiplier (one can show that 4 λε ≈ ε− n−2 ). 4. In general the non-differentiability of F allows for free boundary problems. The easiest example is again the case of the capacity, whose EulerLagrange equation is the so-called Bernoulli free boundary problem: look for an open set A and a function u which is the weak solution of ⎧ ∆u = 0 in Ω \ A ⎪ ⎪ ⎪ ⎨u = 1 on ∂A ⎪ u = 0 on ∂Ω ⎪ ⎪ ⎩ ∂u on ∂A, ∂ν = qε where qε goes to infinity as ε → 0 and play the role of the Lagrange multiplier. Another classical example covered by the problem is the socalled plasma problem (see e.g. [16]).

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3.2 Generalized Sobolev inequality Let us see first some general facts related to the scaling properties of our problem. Denote by S F := S1F (Rn ); i.e., F 1,2 n S = sup F (u) dx : u ∈ D (R ) , |∇u|2 dx ≤ 1 . (17) Rn

Rn

Lemma 4 1. SεF (Ω) ≤ S F for every ε > 0 and for every Ω ⊆ Rn ; 2. The following Generalized Sobolev Inequality holds 22∗ F 2 F (u) dx ≤ S |∇u| dx ∀ u ∈ D1,2 (Ω) ; Ω

3. S

F

Ω

∗

≥ F0 S .

Remark 7. The above lemma is based on the following scaling argument: if u ∈ D1,2 (Ω) and s > 0, define x ∈ D1,2 (sΩ) . us (x) := u s Then we have s n s 2 n−2 F (u ) dx = s F (u) dx and |∇u | dx = s |∇u|2 dx . sΩ

Ω

sΩ −2 n−2 L2

Ω

In particular if we choose s = ∇u then we also obtain sΩ |∇us |2 dx = 1. s Taking u as a competitor for (17), one get the generalized Sobolev inequality. By a cut-off argument the following result can be also proved. Proposition 5 limε→0 SεF = S F . The proof of Lemma 4 and Proposition 5 can be found in [13]. 3.3 Concentration From now on we will consider only the case of Ω bounded and we will write H01 (Ω) instead of D1,2 (Ω). Nevertheless most of the result we will present in this lectures have been obtained for general domains, possibly unbounded. We are interested in the asymptotic behaviour of maximizing sequences for problem (15); i.e., sequences uε such that ∗ F (uε ) dx = SεF + o(1) , (18) ε−2 Ω

and, possibly, in saying something about their optimal profile. The case of F ∈ C 1 (R) has been studied by P.L. Lions [21] as an application of the concentration-compactness principle. The general case has been considered by M. Flucher and S. M¨ uller in ’99 [13] (see also the book of Flucher [10] and the references therein), still in the spirit of to concentrationcompactness.


245

Theorem 6 ([13], Theorem 3) Let uε be a maximizing sequence (i.e. satisfying (18)), then there exists x0 ∈ Ω such that : i) The sequence uε concentrates at x0 in the following sense |∇uε |2 ∗ δx0 ε2

∗

and

|uε |2 ∗ F S δx0 ε 2∗

in M(Ω) ;

(19)

ii) Suppose that S F > max{F0 , F∞ }. Then we identify the optimal profile for the maximizing sequences. Namely there exists a sequence xε converging to 2 x0 such that the sequence wε (x) := uε (xε + ε n−2 x) is compact in D1,2 (Rn ) and every cluster point w is a solution for S F ; i.e., it is a maximum for problem (17). Remark 8. Let us spend a few words about the condition in part ii) of the theorem. This is a natural condition for those who are familiar with the concentration-compactness approach. It guarantees the existence of a ground state and no concentration of optimal sequences for problem (17). In particular it gives a precise rate of concentration for maximizing sequences of the scaled problem SεF (we will see that this condition can be slightly relaxed). It is actually easy to see that S F ≥ max{F0 , F∞ } is always true. It is enough to take an optimal function for S ∗ , e.g. u1 , and define x n . us (x) = s− 2∗ u1 s Then from the generalized Sobolev inequality we obtain S F ≥ F0 S ∗ and S F ≥ F∞ S ∗ taking the limit as s → ∞ and s → 0 respectively. The idea is that the strict inequality, in applying the concentration-compactness principle to the sequence wε , is sufficient to rule out vanishing and concentration. We will see a proof of the concentration result (part i) of Theorem 6) in terms of the variational convergence introduced by De Giorgi in ’75 ([8]), the Γ -convergence. 3.4 Γ + -convergence We are considering maximum problems, so the natural variational convergence is Γ + -convergence. Since in the literature it is mainly Γ − -convergence, the suitable convergence for minimum problems, that is used, we recall quickly the definition and the main properties of Γ + -convergence. In what follows X will be a metric space and τ will denote its topology. Definition 7 Let Fε : X → R, with R = R∪{∞}, be a family of functionals. We say that the sequence Fε Γ + -converges to the functional F : X → R with respect to τ , Γ + (X)

Fε −→ F , if the following two properties are satisfied

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Adriana Garroni τ

i) for every sequence xε → x we have that lim sup Fε (xε ) ≤ F(x) ; ε→0

τ

ii) for every x ∈ X, there exists a sequence xε , such that xε → x and lim inf Fε (xε ) ≥ F(x) ; ε→0

We usually refer to condition i) as the Γ + -limsup inequality and to condition ii) as the existence of a recovery sequence. The idea of Γ + -convergence is that the functional F describes the main features of the sequence Fε in terms of maxima; i.e., it gives the best (maximal) behaviour of the sequences Fε (xε ) along maximizing sequences xε . Remark 9. 1. The Γ + -limit F is upper-semicontinuous with respect to τ . 2. If the topology τ is separable, then Γ + -convergence is compact. 3. If there exists a compact set K ⊆ X such that sup Fε (x) = sup Fε (x) ,

x∈X

x∈K

then a) limε→0 supx∈X Fε (x) = supx∈X F(x) = maxx∈K F(x); b) given a maximizing sequence; i.e., any sequence xε such that lim Fε (xε ) = lim sup Fε (x) ,

ε→0

ε→0 x∈X

any cluster point x of xε is a maximum point for F. A rigorous and complete treatment of the Γ − -convergence for the case of minimum problems, whose definition is perfectly symmetric to the one we gave for Γ + , can be found for instance in [7] or [4]. 3.5 The concentration result in terms of Γ + -convergence It is convenient to modify the functional, rescaling the functions by ε and define ⎧ ⎨ε−2∗ F (εu) dx if |∇u|2 dx ≤ 1 Fε (u) = (20) Ω Ω ∗ ⎩ 0 otherwise in L2 (Ω). The idea is to consider the functionals Fε and find a suitable topology and a limit functional in order to capture the behaviour stated in Theorem 6 by Flucher and M¨ uller.


247

Main requirements: 1) Choose a space X, to which possibly extend the functional Fε , rich enough to give information on the maxima; 2) Find a topology which is compact on X in order to get convergence of maxima. In order to clarify our future choice let us go back to the approach of P.L. Lions. Let uε be a sequence satisfying Ω |∇uε |2 dx ≤ 1. Define µε = |∇uε |2 dx ∗ and try the limit of ε−2 F (εuε ) dx in terms of the limit of µε . to describe Since Ω |∇uε |2 dx ≤ 1, we can always assume that uε converges weakly in H01 (Ω) to some function u and that µε converges weakly∗ in M(Ω) to some measure µ. As above we may decompose µ as follows µi δxi + µ %, (21) µ = |∇u|2 dx + i∈J

% is the non-atomic where µi denotes the positive weight of the atom xi and µ part of µ − |∇u|2 dx, and define ∗

νε = ε−2 F (εuε ) dx . By the generalized Sobolev inequality we have νε (Ω) ≤ S F µε (Ω) and thus we have that up to a subsequence νε converges weakly∗ in M(Ω) to some measure ν. On the other hand by Lemma 1 we know that ∗ ∗ ∗ νi∗ δxi ; νε∗ = |uε |2 dx ν ∗ = |u|2 dx + i∈J

so that by assumption i) on F we get ν ≤ c ν∗ .

(22)

This implies that there exists a function g ∈ L1 (Ω) and non-negative numbers νi such that ν = g(x) dx + νi δxi . (23) i∈J ∗

Then it is clear that, in order to describe the behaviour of ε−2 F (εuε ) dx the weak limit in H01 (Ω) of uε is not enough, but we need also to take into account “how” it converges weakly, which is detected by the weak∗ limit of the measures µε . This suggests the choice of the space X(Ω) as X(Ω) ⊂ H01 (Ω) × M(Ω)

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and precisely, in view of the constraint

Ω

|∇u|2 dx ≤ 1, we will choose

" ! X(Ω) = (u, µ) ∈ H01 (Ω) × M(Ω) : µ ≥ |∇u|2 dx , µ(Ω) ≤ 1 . Consequently the topology τ will be chosen such that ∗ uε u in w − L2 (Ω) τ (uε , µε ) → (u, µ) ⇐⇒ ∗ in M(Ω). µε µ

(24)

(25)

Remark 10. 1. With this choice of the topology the space X(Ω) is metric and separable. 2. It is possible to show that all the pairs (u, µ) ∈ X(Ω) can be obtained as a limit with respect to τ of pairs of the form (uε , |∇uε |2 ). 3. The topology τ is compact in X(Ω), then the Γ + -convergence of functionals in this space implies the convergence of maxima. Now that we have set the right space we have to extend our functional to X(Ω) and this is done, by a little abuse of notation, in the natural way as follows ⎧ ∗ −2 ⎪ F (εu) dx if µ = |∇u|2 dx, ε ⎪ ⎨ Ω Ω Fε (u, µ) = (26) ⎪ ⎪ ⎩ 0 otherwise in X(Ω). Then we look for a functional F which is the Γ + -limit of Fε in X(Ω). Clearly this Γ + -limit has to take into account both, the absolutely continuous part of µ and its atomic part. Theorem 8 ([2], Theorem 3.1) There exists a functional F : X(Ω) → R which is the Γ + -limit of Fε in X(Ω) and it is given by ∗ 2∗ |u|2 dx + S F (µi ) 2 . (27) F(u, µ) := F0 Ω

i∈J

Remark 11. By the Γ + -convergence result we immediately deduce the concentration result. Indeed, as already observed, we have that sup

Fε (u, µ) = SεF →

(u,µ)∈X(Ω)

max

(u,µ)∈X(Ω)

F(u, µ) ;

and hence, since by Proposition 5 we have that SεF → S F , we get S F = max(u,µ)∈X(Ω) F(u, µ). On the other hand by the Sobolev inequality, Lemma 4 and the convexity of the function |t|

2∗ 2

we get


∗

F(u, µ) = F0

|u|2 dx + S F Ω

(Sobolev inequality) ≤ F S ∗ 0 (S ∗ F0 ≤S F )

≤S

(µi )

2∗ 2

i∈J

|∇u|2 dx

2

249

22∗

+ SF

Ω

|∇u| dx

F

2

2∗ (µi ) 2 i∈J

22∗ +

Ω

(µi )

(28)

3 2∗ 2

i∈J

≤ S F µ(Ω) ≤ S F . Since the Sobolev constant is not attained in Ω the first inequality is strict unless u = 0 and the third inequality is strict unless µ = δx0 for some x0 ∈ Ω. In other words F(u, µ) = S F = max F X(Ω)

⇐⇒

(u, µ) = (0, δx0 ) ,

which correspond to the concentration of the maximizing sequence at x0 . Proof (Theorem 8). We give a detailed proof of the Γ + -limsup inequality (condition i) of Definition 7); i.e. we will prove that for every sequence τ (uε , µε ) → (u, µ) then lim sup Fε (uε , µε ) ≤ F(u, µ) .

(29)

ε→0

We can assume (uε , µε ) = (uε , |∇uε |2 ) otherwise the proof is trivial. We already know that in this case ∗ g dx + νi = ν(Ω). (30) Fε (uε , µε ) = νε (Ω) = ε−2 F (εuε ) dx → Ω

Ω

i∈J

We will give the prove in several steps. Step 1. (Localization of the generalized Sobolev inequality) For every δ > 0 there exists a constant k(δ) > 0 such that if 0 < r < R with Rr ≤ k(δ), then for every x0 ∈ Ω

F (u) dx ≤ S F Br (x0 )

22∗

|∇u|2 dx + δ

BR (x)

|∇u|2 dx

(31)

Ω

for every u ∈ H01 (Ω). In order to see this step let define the function log |x0 − x| − log R ,1 (x) = max ϕR r log r − log R

.

This function (the n-capacitary potential of the ball Br with respect to BR ) has the following property

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Adriana Garroni

n |∇ϕR r | dx → 0

as

BR (x0 )

r → 0. R

Moreover it is a cut-off function between Br (x0 ) and BR (x0 ); i.e., ϕR r (x) = 1 n (x) = 0 in R \ B (x ). in Br (x0 ) and ϕR R 0 r Now by H¨ older’s inequality and the Sobolev inequality, for any β > 0 and for any u ∈ H01 (Ω), we get 2 |∇(ϕR r u)| dx BR (x0 )

1 R 2 2 |∇ϕr | |u| dx + (1 + β) |∇u|2 dx ≤ 1+ β BR (x0 ) BR (x0 ) n2 n−2 2 ∗ 1 n |∇ϕR |u|2 dx ≤ 1+ r | dx β BR (x0 ) BR (x0 ) +(1 + β) |∇u|2 dx BR (x0 )

⎛

1 ≤ |∇u|2 dx + ⎝β + 1 + β BR (x0 )

S∗

3 n2 ⎞ n ⎠ |∇ϕR |∇u|2 dx . r | dx

2 BR (x0 )

Ω

Now if β ≤ δ/2 and the ratio between r and R is small enough we get R 2 2 |∇(ϕr u)| dx ≤ |∇u| dx + δ |∇u|2 dx. BR (x0 )

BR (x0 )

Ω

Then the conclusion follows applying the Generalized Sobolev inequality to the function ϕR r u. Step 2. We now prove that νi ≤ S F (µi )

2∗ 2

.

(32)

For any δ > 0, for any atom xi ∈ Ω, with i ∈ J, and Rr ≤ k(δ) we may apply the localization of the generalized Sobolev inequality (31) to the functions εuε and recalling that Ω |∇uε |2 dx ≤ 1 we get ε−2

∗

F (εuε ) dx ≤ S F Br (xi )

|∇uε |2 dx + δ

.

BR (xi )

Taking the limit as ε → 0 we have ν(Br (xi )) ≤ S F (µ(BR (xi )) + δ)

2∗ 2

,

thus the conclusion follows taking the limit as r → 0, by the arbitrariness of R and δ.


251

Step 3.We finally prove that ∗

g ≤ F0 |u|2 .

(33)

Here the idea in that large values of uε do not contribute to the absolutely continuous part of ν. Fix an open subset U of Ω and δ > 0. Let tδ > 0 be such that F (t) ≤ (F0 + δ)|t|2

∗

|t| < tδ . (34) −2∗ Since U is open we have that ν(U ) ≤ lim inf ε→0 U ε F (εuε ) dx and that ∗ ν ∗ (U ) ≥ lim supε→0 U |uε |2 dx; hence, we get ∗ g dx ≤ ν(U ) ≤ lim inf ε−2 F (εuε ) dx ε→0 U U ∗ ≤ lim sup ε−2 F (εuε ) dx ε→0 U ∩{|εuε | ε−2 + H(x0 , y) 1 $ 5 n−2 4 γn = |x0 − y| < −2 ε + H(x0 , y) 1 $ 5 n−2 4 2 γn n−2 . = |x0 − y| < ε 1 + ε2 (H(x0 , x0 ) + o(1)) n

This, recalling that S V = ωn γnn−2 , implies that n 5 n−2 1 |Aε | = S ε 1 + ε2 H(x0 , x0 ) + o(ε2 ) 4 5 n 2 V 2∗ 2 ε H(x0 , x0 ) + o(ε ) . 1− =S ε n−2

V 2∗

4

Remark 12. In the computation above, the quantity which determines the volume of Aε , when x0 varies in Ω is the regular part of the Green’s function, more precisely HΩ (x0 , x0 ).


255

Definition 10 The diagonal of the regular part of the Green’s function is called the Robin function of Ω; i.e., τΩ (x) := HΩ (x, x) . We call the harmonic radius of Ω at x, the positive number ρΩ (x) such that K(ρΩ (x)) = τΩ (x) . The points of Ω where τ (x) attains its minimum (the maxima for ρΩ (x)) are called the harmonic centers of Ω. Remark 13. The computation that we did before shows that among all the super-level sets of the Green’s function, with given capacity, the biggest are those corresponding to the singularity in the harmonic centers of Ω; and hence which concentrates at this points. In the case of a ball BR = BR (0) the Robin function and the harmonic radius can be computed explicitly and they are given by 2−n |x|2 |x|2 . and ρBR (x) = R − τBR (x) = γn R − R R Hence τBR attains its minimum at the origin and tends to infinity when x approaches the boundary. Note that in general the harmonic radius of a domain Ω at a point x is the radius of the ball whose corresponding Robin function evaluated at the origin agrees with τΩ (x). Remark 14. 1. In general, if the boundary of Ω is regular then the Robin function τΩ (x) tends to infinity as x approaches the boundary. Indeed in this case the regular part of the Green’s function, HΩ (x, ·), attains the boundary condition continuously. We will see that we can extend all these notions to the case of general domains (possibly irregular) and that in some cases this property may be false. 2. In the case of the ball the harmonic center is unique and coincides with the center of the ball. It has been proved by Cardaliaguet and Tahraoni [6] that for any bounded convex domain there exists only one harmonic center. 3. The role of the Green’s function for similar problems involving the critical Sobolev exponent was first conjectured by Brezis and Pelletier [5]. In particular the conjecture says that the solutions of the following problem ⎧ p−1 ⎪ in Ω ⎨−∆u = u (41) u=0 on ∂Ω ⎪ ⎩ u>0 in Ω ,

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Adriana Garroni

with p < 2∗ , concentrate at critical points of the Robin function of Ω as p → 2∗ . Namely the points xp where a solution up of problem (41) attains the maximum, converge to a critical point of τΩ (x). This conjecture has been then proved by Rey [22] and Han [18] independently. Later, Flucher and Wei [15] proved that the variational solutions; i.e., the maximizers of ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ p ⎪ ⎪ |u| dx ⎨ ⎬ 1 Ω : u ∈ H (Ω) u = 0 Sp (Ω) = sup , p 2 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ ⎭ |∇u| dx Ω

concentrate at the harmonic center of Ω. In view of the example of the capacity, we expect for our general variational problem SεF (Ω) a result similar to that described in Remark 14 (3). We know that for any maximizing sequence uε , namely Fε (uε ) = SεF (Ω) + o(1), there exist a subsequence (still denoted by uε ) and a point x0 ∈ Ω such that uε concentrates at x0 . On the other hand for any x0 ∈ Ω we can construct a maximizing sequence which concentrates at x0 . Now the question is the following: if we look at maximizing sequences which converge faster, is the concentration point determined by the shape of the domain Ω? This problem has been considered in [11] by means of an asymptotic expansion of the energy. The same result can be stated in terms of Γ + -convergence (see [2]). In the latter approach a way to select among maximizing sequences is to consider a first order expansion in Γ + -convergence (a sort of Taylor expansion in Γ + -convergence). 4.2 Asymptotic expansion in Γ + -convergence Assume that a sequence of functionals Fε , defined in a metric space X, Γ + converges with respect to the topology τ to some functional F : X → R. Suppose now that, following an ansatz, we know that sup Fε = max F + O(λε ) X

X

for some λε = o(1). We then consider the following functional Fε1 (x) =

Fε − max F X

λε

.

If the Γ + -limit F 1 of Fε1 exists, then it clearly will be finite at most on the maxima of F. Furthermore under the usual compactness condition for the topology, we deduce that sup Fε − max F lim sup Fε1 ε→0 X

= lim

ε→0

X

X

λε

= max F 1 , X

(42)


257

which gives an asymptotic expansion for the suprema of Fε ; i.e., sup Fε = max F + λε max F 1 + o(λε ) . X

X

X

(43)

In addition we have that any maximizing sequence xε for Fε1 ; i.e., such that Fε1 (xε ) = sup Fε1 + o(1) , X

converges, up to a subsequence, to a maximum of F 1 . In particular such a maximizing sequence will also satisfy Fε (xε ) = sup Fε + o(λε ) .

(44)

X

In this sense the first order Γ + -limit selects, among all maximizing sequences for Fε , those which converge faster. 4.3 The result In our case the scaling suggested by the computation for the super-level set of the Green’s function is λε = ε2 . Thus our next goal is to compute the Γ + -limit of the following functionals Fε1 (u, µ) =

Fε (u, µ) − S F . ε2

(45)

In order to obtain a non trivial limit for Fε1 we have to assume an additional condition. Indeed we cannot aspect in general a precise rate of convergence ∗ of maximizing sequences. For instance in the critical case F (t) = |t|2 , we may have optimal sequences converging to S ∗ with any rate. We then need an assumption which forbids this scaling invariance effect. As we already briefly mentioned a sufficient condition is given by S F > max{F0 , F∞ }S ∗ .

(46)

Under this condition we know that there exists a ground state w for S F . The qualitative behaviour of such a solution has been stated by Flucher and M¨ uller in [12]. Among other things they prove that there exists a point x0 ∈ Rn and a ball BR0 (x0 ) such that w is radial around x0 outside this ball. More precisely they find a constant W∞ such that w(x) = W∞ K(|x − x0 |)(1 + o(r−2 ))

∀ x : r = |x − x0 | ≥ R0 ;

(47)

in other words w is asymptotically proportional to the fundamental solution. The constant W∞ is given by 2(n − 1) F (w(x)) 2 dx. W∞ = n SF Rn K(|x|)

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Adriana Garroni

Set now w∞ > 0 such that 2 w∞

2(n − 1) F (wk (x)) dx := inf lim inf k→∞ n SF K(|x|) n R

,

(48)

where the infimum is taken among radial maximizing sequences for S F . By using the concentration compactness principle it is shown in [11] that condition (46) implies 0 < w∞ < +∞ .

(49)

We now are ready to state the theorem on the identification of the concentration point in terms of Γ + -convergence. Theorem 11 ([11] and [2]) Under condition (46) we have that there exists the Γ + -limit F 1 of Fε1 defined by (45) in the space X(Ω) = H01 (Ω) × M(Ω) and it is given by ⎧ n F ⎪ if (u, µ) = (0, δx0 ) ⎪ ⎨− n − 2 S w∞ τΩ (x0 ) 1 F (u, µ) = ⎪ ⎪ ⎩−∞ otherwise. As a consequence of this Γ + -convergence result we can deduce the result of concentration for sequences of almost maximizers; i.e., satisfying ∗ F (εuε ) dx = SεF (Ω) + o(ε2 ) . (50) ε−2 Ω

Theorem 12 Under condition (46), all sequences of almost maximizers, in the sense of (50), up to a subsequence, concentrate at a harmonic center of Ω. Proof. Theorem 11 implies that the sequence uε , being a maximizing sequence for Fε1 , up to a subsequence, must concentrate at a point x0 ∈ Ω which satisfies F 1 (0, δx0 ) = max F 1 (u, µ) , X(Ω)

and this implies that τΩ (x0 ) = minΩ τΩ ; hence we deduce the concentration at a harmonic center of Ω. In order to obtain the result above, condition (46) can be relaxed assuming directly condition (49). More details about the importance of condition (49) can be found in [11]. Here we will consider in detail only the proof of Theorem 11 in the case of the Volume problem in (14). In this case we already saw that the solution for S V is given by w(x) = K(|x|) ∧ 1 , and hence w∞ = 1. In the following we will prove Theorem 11 in the capacity case. Namely we will prove


259

(i) (Existence of the recovery sequence) For every x0 ∈ Ω there exists a sequence of sets Aε which concentrates at x0 such that cap(Aε , Ω) = ε2 and ∗ ε−2 |Aε | − S V n τΩ (x0 ) ≥ −S V lim inf 2 ε→0 ε n−2 (ii) (Γ + -limsup inequality) For every sequence of sets Aε which concentrates at some x0 and satisfies cap(Aε , Ω) ≤ ε2 , we have ∗

lim sup ε→0

ε−2 |Aε | − S V n τΩ (x0 ) . ≤ −S V ε2 n−2

Here with concentration of the sets Aε at x0 , we mean |∇uε |2 ∗ δx0 , ε2 where uε denotes the capacitary potential of Aε , or equivalently the pair (uε /ε, |∇uε |2 /ε2 ) converges to (0, δx0 ) in the topology of X(Ω). Proof (i). The existence of the recovery sequence essentially has been already proved when we computed the volume of the super-level sets of the Green’s function. Thus the recovery sequence is given by Aε = {GxΩ0 > ε−2 } and we have ∗ n τΩ (x0 )ε2 + o(ε2 ) . |Aε | ≥ ε2 S V 1 − n−2 The proof of the Γ + -limsup inequality is based on an asymptotic formula for the capacity of the small sets. The crucial lemma is the following. Lemma 13 Let x0 ∈ Ω and let Aε be a sequence of subsets of Ω, with |Aε | > 0, such that |∇uε |2 ∗ δx0 , cap(Aε , Ω) where uε is the corresponding capacitary potential, then lim inf ε→0

1 1 + ≥ τΩ (x0 ) . ∗ n cap(Aε , R ) cap(Aε , Ω)

(51)

Remark 15. The assumption of concentration for the sets Aε given in the lemma above can be given in the following weaker form χAε ∗ δx0 |Aε | (which is actually what we need for the proof of the theorem in the general case). A complete proof of Lemma 13 can be found in [11], Lemma 16. Let us see now how we obtain the proof of the Γ + -limsup inequality using the lemma.

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Proof. (ii) Without loss of generality we may assume that Aε satisfies lim sup ε→0

cap(Aε , Ω) = 1. ε2

Otherwise the sequence Aε will not be a maximizing sequence and the Γ + limsup would be trivially satisfied. Then we can apply Lemma 13. By symmetrization we have 1 |Aε | n ∗ with ρε = |Aε | = |Bρε | = |Aε | ωn and hence cap(A∗ε , Rn )

1 1 = = K(ρε ) γn

|Aε | ωn

n−2 n

=

|Aε | SV

n−2 n .

Thus Lemma 13 implies

SV |Aε |

n−2 n

−

1 ≥ τΩ (x0 ) + o(1) . ε2

Then SV |Aε | = SV n ∗ ≤ 2 ε [1 + (τΩ (x0 ) + o(1))ε2 ] n−2

1−

n (τΩ (x0 ) + o(1))ε2 n−2

,

which gives the Γ + -limisup inequality. In the proof of the identification of concentration points for the case of the Volume Functional, Lemma 13 plays a crucial role. It measures the contribution of the boundary of Ω in the computation of the capacity with respect to Ω. We will not give the proof of Theorem 11 in the general case. The proof of the Γ + -limsup inequality uses similar ideas, but it requires a fine analysis of the asymptotic behaviour of a maximizing sequence and the corresponding super-level sets. We just give a few hints for the construction of the recovery sequence for the general case. This indeed gives us the occasion to recall a rearrangement technique which is one of the main tools to get lower bounds for this kind of problems. This is the classical technique of harmonic transplantation. Introduced by Hersch in ’69 ([20]) it provides information somehow complementary to the information obtained by radial rearrangement. 4.4 Harmonic transplantation Definition 14 Denote by G0B the Green’s function of the ball B = BR (0) with singularity at 0. Given an arbitrary radial function


261

U : BR (0) → R we can write it as U = ϕ ◦ G0B , for a suitable real function ϕ. Now fix x0 ∈ Ω. The harmonic transplantation of U from (BR (0), 0) to (Ω, x0 ) is the function u = ϕ ◦ GxΩ0 : Ω → R . Theorem 15 The harmonic transplantation u of U from (BR (0), 0) to (Ω, x0 ) satisfies 1. The Dirichlet integral is preserved; i.e., |∇u|2 dx = Ω

|∇U |2 dx ;

BR (0)

2. If R = ρ(x0 ) is the harmonic radius of x0 in Ω, then F (u) dx ≥ F (U ) dx Ω

BR (0)

for every non-negative function F ; 3. If Uε is a sequence of radial functions, with Uε = ϕε ◦ G0B , which concen∗ trate at 0 in the sense that |∇Uε |2 δ0 , then the corresponding harmonic x0 transplantation uε = ϕ ◦ GΩ from (BR , 0) to (Ω, x0 ) are concentrates at x0 . Proof. Part 1 is a consequence of the co-area formula. Indeed, by using GxΩ0 ∧t as test function in problem (37) and integrating by parts, it can be easily seen that |∇GxΩ0 | dHn−1 = 1 ∀ t>0 ∀ Ω. x

{GΩ0 =t}

Then, using the level sets of the Green’s function and recalling that u = ϕ◦GxΩ0 and U = ϕ ◦ G0B , we can write 2 |∇u| dx = |ϕ (GxΩ0 )|2 |∇GxΩ0 |2 dx Ω Ω +∞ +∞ = |ϕ (t)|2 |∇GxΩ0 | dHn−1 dt = |ϕ (t)|2 dt , 0

x

{GΩ0 =t}

0

independently of Ω. Part 2 is a consequence of the following Mean Value Inequality: Fix x0 ∈ Rn , t > 0 and ρ > 0; then among all Ω such that x0 ∈ Ω and ρ = ρΩ (x0 ) the following quantity 1 x0 dx x0 ∂{GΩ >t} |∇GΩ |

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is minimal for Ω = Bρ (x0 ). This fact can be proved using the isoperimetric inequality on the level sets of the Green’s’ function and the properties of the harmonic radius (see [3] for a detailed proof). Using this result we then have

+∞

F (u) dx =

F (ϕ(t))

x

∂{GΩ0 >t}

0

Ω

1 dx dt ≥ |∇GxΩ0 |

F (u) dx. Bρ(x0 )

As we said harmonic transplantation is the basic tool in order to construct a recovery sequence in the general case. To understand how to make use of this tool, let us briefly show the main lines in the construction. Suppose that we are able to construct a recovery sequence in the special case Ω = BR (0) with concentration at the center; namely suppose that we have a sequence → R of radial functions which concentrates at the origin, with U ε : BR (0) 2 |∇U | dx = 1 and satisfying ε BR ε lim inf ε→0

−2∗

F (εUε ) dx − S F BR (0)

≥−

ε2

n w∞ τBR (0) . n−2

(52)

Now given Ω ⊆ Rn , x0 ∈ Ω and fix R = ρΩ (x0 ) the harmonic radius of Ω at x0 , by harmonic transplantation we get a sequence uε : Ω → R, which concentrates at x0 , such that |∇uε |2 dx = 1, Ω

and from the form of the Robin function for a ball, the definition of the harmonic radius and (52) satisfies ∗ ∗ ε−2 F (εUε ) dx − S F ε−2 F (εuε ) dx − S F B (0) R Ω ≥ lim inf lim inf ε→0 ε→0 ε2 ε2 n w∞ τBR (0) ≥− n−2 n w∞ K(ρ(x0 )) =− n−2 n w∞ τΩ (x0 ) ; =− n−2 hence (uε , |∇uε |2 ) is a recovery sequence for (0, δx0 ). Remark 16. The case Ω = BR (0), with concentration in the center can be done by hand, taking into account that the solution w for S F is radial and behaves as the fundamental solution at ∞. Then one can construct Uε truncating w and scaling it (see [11] for details).


263

5 Irregular domains In this last section we will briefly consider the case of domains Ω with possibly irregular boundary. In order to deal with this case we should be able to define the Robin function and the harmonic radius up to the boundary for possibly irregular domains. Indeed if the domain is irregular in general one can not aspect the harmonic center being attained at an interior point (see Example 3). With the following example we exhibit a domain whose harmonic center is at the boundary. Example 3. Let Ω0 = B1 (0) and let τΩ0 be the corresponding Robin function. The harmonic center for Ω0 is 0 and τΩ0 is strictly convex. The idea is to construct two symmetric sequences of small balls centered in the points ( 21n , 0, . . . , 0) and (− 21n , 0, . . . , 0) respectively with radii which go to zero, in a way that the set obtained from Ω0 by subtracting a finite number of symmetric pairs of balls has its unique harmonic center in the origin.

+ − Fig. 1. The set Ωn = Ω0 \ (∪n i=1 (Bρi (xi ) ∪ Bρi (xi )))

1 Let us denote by x± n = (± 2n , 0, . . . , 0), n ∈ IN and let ε1 > 0 be such that 0 < ε1 < minΩ0 \B 1 (0) τΩ0 − minB 1 (0) τΩ0 . Fix 0 < α < 1/2, let ρ1 > 0 and 4

4

− denote Ω1 = Ω0 \ (Bρ1 (x+ 1 ) ∪ Bρ1 (x1 )). It is easy to check that τΩ1 converges − α uniformly to τΩ0 , as ρ1 tends to zero, in Ω0 \ (Bρα1 (x+ 1 ) ∪ Bρ1 (x1 )) and the same is true for the derivatives. Thus we can choose ρ1 small enough such that − ± ± α α τΩ1 is strictly convex on Ω0 \ (Bρα1 (x+ 1 ) ∪ Bρ1 (x1 )), Bρ1 (x1 ) ∩ B 14 (x1 ) = ∅ and we have ε1 − ∀ x ∈ Ω0 \ (Bρ1 (x+ τΩ0 (x) ≤ τΩ1 (x) ≤ τΩ0 (x) + 1 ) ∪ Bρ1 (x1 )) . 2 This implies that the harmonic center of Ω1 is unique, and arguing by symmetry, we conclude that it is in the origin. By induction we can construct a sequences {ρn }, such that the sets Ωn = − Ω0 \ (∪ni=1 (Bρi (x+ i ) ∪ Bρi (xi ))) have a unique harmonic center at the origin. In particular dist(0, ∂Ωn ) → 0 as n → ∞.

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Question: what happens in domains with irregular boundary? Can we still prove a concentration result like Theorem 4.3? Note that all the techniques we used in the previous section make use of the fact that the concentration point is an interior point for Ω. Nevertheless the answer to the second question is yes, in order to state and prove the concentration result we need a good definition of the Robin function for irregular domains up to the boundary. This can be done in three steps. Step 1. For any x0 ∈ Ω we can define the regular part of the Green’s function HΩ (x0 , ·) as the solution in the sense of Perron-Wiener-Brelot of the following Dirichlet problem ⎧ ⎪ ⎨ ∆y HΩ (x0 , y) = 0 in Ω, (53) ⎪ ⎩ HΩ (x0 , y) = K(|x0 − y|) on ∂Ω; i.e., HΩ (x0 , ·) is the infimum of all superharmonic functions u such that lim inf u(z) ≥ K(|x0 − y|) z→y z∈Ω

for every y ∈ ∂Ω (see [19]). Note that K(|x0 − ·|) is an admissible boundary condition in order to get a unique solution for problem (53). Step 2. We may extend H(x0 , ·) to the boundary of Ω as follows % Ω (x0 , y0 ) = lim inf H(x0 , y) . H y→y 0

y∈Ω

Step 3. We now can define the Robin function up to the boundary as % Ω (x0 , x0 ) . τΩ (x0 ) = H

With the definition above of τΩ we can state and prove Theorem 4.3 for any domain, possibly irregular. Remark 17. One could be tempted to define the Robin function up to the boundary simply taking the lower semi-continuous extension of τΩ (x) = HΩ (x, x) with x ∈ Ω. In dimension n ≥ 5 one can construct an example which shows that the two procedures do not give the same function; i.e., the Robin function defined by Step 3 can be strictly smaller at the boundary than its lower semi-continuous extension from Ω. The definition of τΩ permits also to show that it satisfies the following properties.


265

Proposition 16 a) τΩ is lower semi-continuous in Ω; b) For any ρ > 0 and any x0 ∈ Ω we denote by τρ the Robin function corresponding to the domain Ω ∪ Bρ (x0 ). Then we have that τρ converges increasingly to τΩ as ρ → 0. Remark 18. Note that property b) in the proposition above is essential to extend the result to arbitrary domains. The idea is that it permits to consider a boundary point of Ω as an interior point for a slightly perturbed domain and hence use the same techniques used in the regular case. Indeed first it can be shown that with this definition of τΩ , harmonic transplantation works up to the boundary. Second, it permits to extend Lemma 13 for sets which concentrates at points x0 of the boundary where τΩ (x0 ) < ∞. In fact the lemma does not require regularity, but it require x0 to be an interior point. Then it can be applied to the set Ω ∪ Bρ (x0 ) and gives lim inf ε→0

1 1 + ≥ τρ (x0 ) . ∗ n cap(Aε , R ) cap(Aε , Ω ∪ Bρ (x0 ))

By the fact that cap(Aε , Ω ∪ Bρ (x0 )) < cap(Aε , Ω) and the previous proposition we get 1 1 + ≥ τΩ (x0 ) . lim inf ∗ n ε→0 cap(Aε , R ) cap(Aε , Ω) Remark 19. Proposition 16 shows that τΩn (x) constructed in Example 3 converges to the Robin function τΩ∞ (x) for the set + − Ω∞ = Ω0 \ (∪∞ i=1 (Bρi (xi ) ∪ Bρi (xi ))) ,

for every x ∈ Ω ∞ . In particular, since {τΩn } is an increasing sequence, 0 is the harmonic center of Ω∞ and by construction belongs to the boundary of Ω∞ .

References 1. Alberti, G., Baldo, S., and Orlandi, G. Variational convergence for functionals of Ginzburg-Landau type. Indiana U. Math. J. 54 (2005), 1411–1472. 2. Amar, M., and Garroni, A. Γ -convergence of concentration problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 151–179. 3. Bandle, C., and Flucher, M. Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations ∆U = eU and ∆U = U (n+2)/(n−2) . SIAM Rev. 38 (1996), 191–238. 4. Braides, A. Γ -convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002. 5. Brézis, H., and Peletier, L. A. Asymptotics for elliptic equations involving critical growth. In Partial differential equations and the calculus of variations, Vol. I, vol. 1 of Progr. Nonlinear Differential Equations Appl. Birkh¨ auser Boston, Boston, MA, 1989, 149–192.

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6. Cardaliaguet, P., and Tahraoui, R. On the strict concavity of the harmonic radius in dimension n ≥ 3. J. Math. Pures Appl. 81 (2002), 223–240. 7. Dal Maso, G. An introduction to Γ -convergence. Birkh¨ auser, Boston, MA, 1993. 8. De Giorgi, E., and Franzoni, T. Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 58 (1975), 842–850. 9. Flucher, M. An asymptotic formula for the minimal capacity among sets of equal area. Calc. Var. Partial Differential Equations 1 (1993), 71–86. 10. Flucher, M. Variational problems with concentration. Birkh¨ auser Verlag, Basel, 1999. 11. Flucher, M., Garroni, A., and M¨ uller, S. Concentration of low energy extremals: Identification of concentration points. Calc. Var. Partial Differential Equations 14 (2002), 483–516. 12. Flucher, M., and M¨ uller, S. Radial symmetry and decay rate of variational ground states in the zero mass case. SIAM J. Math. Anal. 29 (1998), 712–719. 13. Flucher, M., and M¨ uller, S. Concentration of low energy extremals. Ann. Inst. H. Poincaré Anal. Non Lin´ eaire 16 (1999), 269–298. 14. Flucher, M., and Rumpf, M. Bernoulli’s free-boundary problem, qualitative theory and numerical approximation. J. Reine Angew. Math. 486 (1997), 165–204. 15. Flucher, M., and Wei, J. Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots. Manuscripta Math. 94 (1997), 337–346. 16. Flucher, M., and Wei, J. Asymptotic shape and location of small cores in elliptic free-boundary problems. Math. Z. 228 (1998), 683–703. 17. Frehse, J. Capacity methods in the theory of partial differential equations. Proc. Amer. Math. Soc. 84 (1982), 1–44. 18. Han, Z.-C. Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 159–174. 19. Helms, L. Introduction to potential theory. John Wiley & Sons Inc., New York, 1969. 20. Hersch, J. Transplantation harmonique, transplantation par modules, et théorèmes isopérimétriques. Comment. Math. Helv. 44 (1969), 354–366. 21. Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1 (1985), 45–121. 22. Rey, O. Proof of two conjectures of H. Brézis and L. A. Peletier. Manuscripta Math. 65 (1989), 19–37. 23. Talenti, G. Elliptic equations and rearrangements. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), 697–718.

Gamma-convergence of gradient flows and applications to Ginzburg-Landau vortex dynamics Sylvia Serfaty Courant Institute of Mathematical Sciences 251 Mercer St, New York NY 10012, USA e-mail: [email protected], http://www.math.nyu.edu/faculty/serfaty Summary. We present in parallel an abstract method of Γ -convergence of gradient flows, designed to pass to the limit in PDEs which are steepest-descent for functionals which have an asymptotic Γ -limit energy; together with the application to the Ginzburg-Landau energy. We give schematic proofs of the Γ -convergence results for Ginzburg-Landau and of the derivation of the dynamical law of vortices through the abstract method.

1 Introduction 1.1 Presentation of the Ginzburg-Landau model The Ginzburg-Landau energy was introduced by Ginzburg and Landau in the 50s as a model for superconductivity. It was first a phenomenological theory, but it was later derived (in a certain limit) from the microscopic (quantic) theory of Bardeen-Cooper-Schrieffer. It is now a widely accepted model, which has earned its inventors the Physics Nobel Prize (to Ginzburg, Abrikosov, and in 2003 Ginzburg). Another motivation is the modelling of superfluidity (a phenomenon very close to superfluidity, both mathematically and physically, with a joint Nobel Prize for Legett in 2003) and of Bose-Einstein condensates in rotation (Bose-Einstein condensates were predicted by Bose and Einstein in the early 20th century, and only first realized experimentally in the 90’s (it was worth another Nobel Prize...). All these physical phenomena have in common the appearance of topological vortices, which are the main object of our study. Superconductors have this striking feature that “they repel an applied magnetic field” (this is called the Meissner effect). This is true at least when the intensity of the applied field hex is not too large; when it becomes larger than a first critical field Hc1 , then the first vortices appear and the magnetic

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Sylvia Serfaty

field penetrates through them; when the applied field is further raised, there are more and more vortices, until superconductivity is totally destroyed and the magnetic field completely penetrates the sample. For further reference, we refer to the physics literature, e.g. [24, 7]. The samples are 3D, however, we will consider only the 2D model for simplicity (it already contains most of the important features). The 2D GinzburgLandau energy in non-dimensional form is 2 1 1 |∇A u|2 + |curl A − hex |2 + 2 1 − |u|2 . (1) Gε (u, A) = 2 Ω 2ε Here Ω denotes a smooth bounded and simply connected domain corresponding to the cross-section of the sample (assuming everything is translationinvariant in the third direction). The function u : Ω → C is called the order parameter, |u(x)|2 ≤ 1 indicates the local (normalized) density of superconducting electrons (the “Cooper pairs”). Where |u(x)| ∼ 1 it is the superconducting phase, where |u(x)| ∼ 0, it is the normal phase. This order parameter is coupled, in a gauge-invariant fashion, to a magnetic potential A : Ω → R2 , and the function h = curl A = ∂2 A1 − ∂1 A2 is the induced magnetic field in the sample. The real parameter hex is the intensity of the external applied magnetic field. The parameter 1/ε is called the Ginzburg-Landau parameter, it is a dimensionless parameter depending on the material (ratio of two characteristic lengths). When 1/ε is large enough, we are in the category of “type-II” superconductors, when ε → 0, they are sometimes called “extreme type-II” (or this is also called the “London limit”). This is the asymptotic regime we will be interested in. Vortices Vortices are objects centered at zeros of the order parameter u which carry a nonzero topological degree. Typically, around a vortex centered at a point 0| ) where f (0) = 0 x0 , u “looks like” u = ρeiϕ with ρ(x0 ) = 0 and ρ = f ( |x−x ε and f tends to 1 as r → +∞, i.e. its characteristic core size is ε, and 1 ∂ϕ =d∈Z 2π ∂τ is an integer, called the degree of the vortex. For example ϕ = dθ where θ is the polar angle centered at x0 yields a vortex of degree d. We have the important relation di δai curl ∇ϕ = 2π i

where the ai ’s are the centers of the vortices and the di their degrees.

Gamma-convergence of gradient flows

269

Simplified model (no magnetic coupling) A simplified model consists in taking A = 0 and hex = 0, then the energy reduces to 1 (1 − |u|2 )2 |∇u|2 + (2) Eε (u) = 2 Ω 2ε2 with still u : Ω → C. Critical points of this energy are solutions of −∆u =

u (1 − |u|2 ). ε2

(3)

The first main study of this functional was done by Bethuel-Brezis-Hélein in the book [2]. Since then, a large literature on it has developed. In these notes, for simplicity, we will focus only on this energy (2), however all our results can be extended to the case of (1). For more reference on (1), and results on its minimizers, their vortices, critical fields, etc, we refer to the monograph [20] and the references therein. In what follows we will often be a little imprecise in the statements for the sake of simplicity, however exact and rigorous corresponding statements can easily be found in the references. 1.2 Gamma-convergence Let us now present the totally independent concept of Γ -convergence. It was introduced by De Giorgi in the 70s, it served to unify various notions of variational convergence. The idea is dimension-reduction: when there is a small parameter ε → 0, reduce the minimization of some original functionals Eε to that of a limiting energy F , defined on a lower-dimensional space. A celebrated example of Γ -convergence was the case of the energy of the “gradient theory of phase-transitions” studied in the 80’s by Modica and Mortola (see also Sternberg): 1 |∇u|2 + (1 − u2 )2 u:Ω→R (4) Mε (u) = ε ε Ω Ω that is the same as (2) but for real-valued functions. It was established that if for a family uε , Mε (uε ) ≤ C, then, up to extraction of a subsequence, uε → u0 in BV (Ω) (the space of functions of bounded variation), with u0 valued in {1, −1} and 8 4 4 Γ 8 |Du0 | = u0 BV Mε −→ per γ = per (∂{u0 = 1}) = 3 3 3 3 where γ (typically a codimension 1 object), is the interface between {u0 = 1} and {u0 = −1}.

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A trick that was used was to write a2 + b2 ≥ 2ab (with equality if a = b) hence 1 u3ε 2 2 2 2 ε |∇uε | + (1 − uε ) ≥ 2 |∇uε ||1 − uε | ≥ 2 ∇ uε − 3 ε Ω Ω Ω and thus passing to the limit, lim Mε (uε ) ≥

ε→0

8 per γ. 3

Conversely, given an interface γ (a curve if Ω ⊂ R2 ), one can construct uε such that Mε (uε ) → 83 per(γ). This necessitates to paste transversally to γ the √ optimal profile such that a = b above i.e. ε|∇u| = √1ε 1 − u2 , that is x 1 uε (x1 , x2 ) tanh ε where x1 is the coordinate in the direction normal to γ. Definition 17 (Γ -convergence) A family of functionals Eε (defined on Mε ) Γ -converges to a functional F (defined on N ) if S

1. If Eε (uε ) ≤ C then up to extraction of a subsequence, uε u ∈ N , and S for every uε u ∈ N we have lim Eε (uε ) ≥ F (u)

ε→0

S

2. For every u ∈ N , there exists uε ∈ Mε u ∈ N such that lim Eε (uε ) ≤ F (u)

ε→0

The sense of convergence S is to be specified beforehand. It can be a weak or strong convergence of uε , it can also be a convergence of a nonlinear function of uε . S In the case of the functional Mε , one should take uε γ ⇐⇒ uε → u0 in 4 n−1 1 #γ where γ denotes a codimension one rectifiable L (Ω) with Du0 = 3 H current, and H the Hausdorff measure. Γ -convergence thus requires two conditions: a lower bound, usually obtained via abstract arguments (together with a compactness result), and an upper bound, usually obtained via explicit constructions. Proposition 1 If Eε Γ -converges to F and uε minimizes Eε with Eε (uε ) ≤ S C, then, up to extraction uε u and u minimizes F . S

Proof. By 1) of the definition, after extraction, uε u and limε→0 Eε (uε ) ≥ F (u). Let us assume that there exists u0 ∈ N such that F (u0 ) < F (u), then by 2) of the definition, there exists vε such that limε→0 Eε (vε ) ≤ F (u0 ) < F (u) ≤ limε→0 Eε (uε ). Thus for ε small enough, we find Eε (vε ) < Eε (uε ) contradicting the minimality of uε . Hence u must minimize F .


271

In other words “minimizers converge to minimizers”. Minimizing Mε defined over H 1 (Ω, R) for example reduces to minimizing F (γ) = per(γ) defined over finite-perimeter sets. It thus achieves a dimension-reduction (since the set of finite-perimeter objects has, somewhat, a lower dimension than H 1 (Ω, R)). In general, not much more can be said. For example uε local minimizer of Eε S does not imply uε u local minimizer; or uε critical point of Eε does not S imply uε u critical point of F . It is easy to construct finite-dimensional counter-examples. 1.3 Γ -convergence of Ginzburg-Landau Energy lower bound To obtain a lower bound (and thus a Γ -convergence result) for the GinzburgLandau functional (2) is more difficult than for Mε (the a2 + b2 ≥ 2ab trick doesn’t work). Let us present (formally) some essential ingredients of the analysis of [2]. What is the cost of a radial vortex of degree d of the form f rε ei dθ ? First, formally 1 2

|∇u|2 ≥ BR

1 2

|f |2 R≥|x|≥ε

d2 r dr dθ = πd2 r2

ε

R

R dr = πd2 log r ε

(5)

where we have assumed that f is close to 1 for |x| ≥ ε. In fact this bound is optimal, at least in the case d = ±1 as can be seen: if u ∈ S1 , u = eiϕ , and |∇u| = |∇ϕ|, so 2 ∂ϕ 1 R 1 2 dr |∇u| ≥ 2 R≥|x|≥ε 2 ε ∂Br ∂τ 2 1 R ∂ϕ 1 ≥ dr (by Cauchy-Schwarz) 2 ε 2πr ∂ Br ∂τ 1 4π 2 R dr R ≥ = π log 2 2π ε r ε valid for any degree ±1 vortex (not necessarily radial). Vortices of degree > 1 cost more energy than several vortices of degree 1 and are in fact unstable. The cost of f rε imposes the length scale ε, and costs only O(1), which is negligible compared to log 1ε . If uε has vortices at points aε1 , · · · , aεn , of degrees d1 , · · · , dn , one expects that 1 |di | log . Eε (uε ) ≥ π ε i

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In fact, this estimate has been made rigorous under certain conditions in [2], and more generally with the “ball construction method” of Sandier/Jerrard (see [17, 11, 20]). How to trace the vortices? The easiest way is to use the current iu, ∇u (or the “superconducting current” iu, ∇A u for the case with magnetic field) where ., . denotes the scalar product in C as identified with R2 , i.e. iu, ∇u = (u × ∂1 u, u × ∂2 u) with × the vector product in R2 . Writing u = ρeiϕ we have (at least formally) iu, ∇u = ρ2 ∇φ and since ρ is close to 1 on length scales ε, the quantity curl iu, ∇u = curl (ρ2 ∇ϕ) curl ∇ϕ = 2π

di δai

(6)

i

can be used to trace the vortices. This is also called the Jacobian determinant if written (with differential forms) Ju = 12 diu, du = 12 idu, du = ux1 × ux2 . The approximation is justified as a limit as ε → 0: there exists a Theorem 1 (see [13, 20]) Assume Eε (u) ≤ C| log ε|, then ri ≤ o(1) as family of disjoint closed balls Bi = B(ai , ri ) with |log ε|−2 ≤ ε → 0, such that 1 ⊂ ∪i Bi (the Bi ’s cover the zeroes of uε ) |u| ≤ 2 ri 1 2 −C di = deg (u, ∂Bi ) (7) |∇uε | ≥ π |di | log i 2 ∪i Bi ε i |di | i curl iuε , ∇uε − 2π di δai (C 0,γ (Ω))∗ ≤ o(1) as ε → 0 (8) 0

i

Combining the upper bound Eε (u) ≤ C|log ε| and the lower bound (7), we deduce that i |di | ≤ C for a constant C independent of ε and thus the number of vortices of nonzero degree remains bounded independently of ε. ε once Thus, if uε is a family of such configurations, the ai ’s are found, we may extract a subsequence such that i di δaεi → i di δai in the weak sense of measures. These fixed points a1 , · · · , an are the limiting vortices. We will sometimes write u = (ai , di ) for the limiting points+degrees configurations in (Ω × Z)n . Then the Γ -convergence result can simply be written Theorem 2 1) Assume uε is such that s

uε u = (ai , di )

in the sense

Eε (uε ) |log ε|

≤ C, then up to extraction

curl iuε , ∇uε 2π

n i=1

and

di δai = 2J


273

n Eε (uε ) ≥π |di | = J ε→0 |log ε| i=1

lim

2) Conversely given any (ai , di ) ∈ (Ω × Z)n , there exists uε such that n ε (uε ) limε→0 E|log i=1 |di | = J . ε| ≤ π This result is not very interesting since it reduces minimizing Eε to minimizing the number of points!... It is mostly interesting in higher dimensions. Then, in 3D for example, vortices are not points but vortex-lines, and the Jacobian Juε = 12 d(iuε , duε ) can be seen as a current carried by the vortex-line, converging to a π times integer-multiplicity dimension 1 rectifiable current (i.e. line) J and Eε Γ −→ J = length of line (or surface...) |log ε| is the lower bound of Γ -convergence (see [13] and Section 3.1). Thus, Γ convergence reduces to minimizing the length of the line, leading to straight lines, a nontrivial problem. In higher dimensions, it leads to codimension 2 minimal currents similarly to Mε (see [16, 3]). In fact, in order for the problem to become interesting in 2D, we need to impose some boundary conditions, for example uε = g ∂Ω with deg g = 0 so that there have to be vortices, and to look at the next order of the energy in the expansion. This rather arbitrary boundary requirement is in contrast with the case of the full functional (1), for which the natural boundary condition is Neumann, and vortices appear due to the applied magnetic field. Renormalized energy Let us return to lower bounds in order to look for the next order term in the energy (still with formal arguments). Cutting out holes ∪i B(ai , ρ) of fixed size ρ around the limiting vortices ai , we may assume that |u| ∼ 1 in Ω\ ∪i B(ai , ρ) = Ωρ , and that u = eiϕ , with ϕ a real-valued function, not singlevalued though (i.e. only defined modulo 2π). Minimizing the energy outside of the holes amounts to solving 1 |∇u|2 . min 1 2 Ωρ u:Ωρ →S u=g on ∂Ω deg(u,∂B(ai ,ρ))=di

This is a harmonic map problem, whose solution is given in terms of ϕ by ⎧ in Ωρ ⎨ ∆ϕ = 0 ∂ϕ given on ∂Ω ⎩ ∂τ ∂ϕ = 2πd . i ∂B(ai ,ρ) ∂τ and in terms of the harmonic conjugate Φ such that ∇ϕ = ∇⊥ Φ, by

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⎧ ∆Φ = 0 in Ωρ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂Φ given on ∂Ω or φ = 0 on ∂Ω for Neumann b.c. ∂n ⎪ ⎪ ⎪ ∂Φ ⎪ ⎪ = 2πdi . ⎩ ∂B(ai ,ρ) ∂n As ρ → 0, Φ behaves like the solution of ∆Φ0 = 2π i di δai Ω ∂Φ0 given or Φ0 = 0 on ∂Ω for Neumann b.c. ∂n Introducing the Green’s kernel associated to Ω (with the right boundary condition), which has a log |x − y| type singularity, and its regular part S(x, y) = 2πG(x, y) − log |x − y|, we have dj G(x, aj ). Φ0 (x) = 2π j

With this relation, Ω |∇Φ0 |2 is infinite but would write formally like |∇Φ0 |2 = −2π di Φ0 (ai ) = −4π 2 di dj G(ai , aj ). Ω

i

i,j

1

1

Now we wish to estimate 2 Ωρ |∇ϕ|2 = 2 Ωρ |∇Φ|2 and it is approximately equal to 1 1 |∇Φ0 |2 π d2i log + Wd (a1 , · · · , an ) + o(1) as ρ → 0 (9) 2 Ωρ ρ i where Wd (a1 , · · · , an ) = −π

i=j

di dj log |ai − aj | − π

S(ai , aj ).

(10)

i,j

The function W was called the renormalized energy in [2]. It contains the (logarithmic) interaction energy between the vortices: we see that vortices with degrees of same sign repel each other while vortices with degrees of opposite signs attract. The d2 log ρ1 term corresponds to the self-interaction, or cost of the vortex of core of size ρ, it is what replaces the infinite term in the formal calculation. Now (9) is a good estimate for the optimal energy outside of the holes, while the energy in holes of size ρ was estimated through (5). Combining these estimates, we are led to the major result of [2]: Theorem 3 ([2]) Assume Eε (uε ) ≤ C|log ε| and uε = g on ∂Ω, with deg g = 0. Then, up to extraction,


curl iuε , ∇uε 2π

n

di δai

275

di ∈ Z

i=1

and Eε (uε ) ≥ π

n

|di | log

i=1

1 + Wd (a1 , . . . , an ) + o(1) ε

as

ε→0

So for a given degree d > 0 on the boundary, in order to minimize the energy, one needs to choose d vortices of degree +1, and then to minimize the remaining interaction term W which is independent of ε and governs the locations of the limiting vortices. From now on, we will reduce to the case di = ±1 and will use Theorem 4 1. Assume Eε (uε ) ≤ C|log ε| and uε = g on ∂Ω or on ∂Ω, then, up to extraction, curl iuε , ∇uε 2π

n

∂uε ∂n

=0

di δai

i=1

and if ∀i, di = ±1, lim Eε (uε ) − πn|log ε| ≥ Wd (a1 , · · · , an ).

ε→0

2. For all (ai , di ), di = ±1, there exists uε such that lim Eε (uε ) − πn|log ε| ≤ Wd (a1 , · · · , an ).

ε→0

Phrased this way, it is a result of Γ -convergence of Eε − πn|log ε|, and the Γ -limit, Wd is nontrivial. We thus reduce minimizing Eε to minimizing Wd which is a finite-dimensional problem (interaction of point charges). Thus we see why it is interesting to study this asymptotic limit ε → 0 because the vortices become point-like and the problem reduces to a finite-dimensional one.

2 The abstract result for Γ -convergence of gradient-flows 2.1 The abstract situation Let Eε be again a family of functionals defined on Mε (see [19] for an idea of what kind of space Mε should be...) and F be a functional defined on N such that Eε Γ -converges to F for the sense of convergence S (in the sense of Definition 1). If we consider a solution of the gradient-flow (or steepest descent) of Eε on Mε i.e.

276

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∂t uε = −∇Eε (uε ), S

does uε (t) u(t) for some u(t) and more importantly, does u(t) satisfy ∂t u = −∇F (u)? An example with a positive answer is that of the functional Mε whose L2 gradient-flow is the Allen-Cahn equation ∂t u = ∆u +

u (1 − |u|2 ) ε2

(11)

Γ

We saw that Mε → F = 83 per (γ). In fact it is true that solutions of the AllenCahn equation converge to interfaces which evolve according to the gradient flow of that perimeter functional F , which is mean-curvature flow. This result is a delicate one, which has been proved with PDE methods (see [8, 6, 9, 10]). Let us point out that the answer is in general negative without further assumptions. Indeed, a necessary condition is that critical points of Eε should converge to critical points of F , but we already mentioned that when Eε Γ -converges to F , this is not necessarily true. We are searching for - an abstract result - an energy-based method - new extra conditions for convergence to occur. Observe that these have to involve the C 1 (or tangent) structure of the energy landscape, i.e. be conditions on the derivatives of the energy and not only of the energies themselves (otherwise it is easy to perturb the energy by a small perturbation in C 0 which adds new critical points which do not converge to critical points). We have been sloppy until now, by writing ∂t uε = −∇Eε (uε ) and calling this the gradient-flow of Eε . Since we are in infinite dimensions (in general), we need to specify what we mean by gradient, i.e. gradient with respect to which structure. There are many possible choices, each leading to a different gradient-flow. For example, the Allen-Cahn equation (11) above is the gradient flow for Mε for the L2 structure. We could consider other structures, such as the gradient-flow with respect to the H −1 structure, it is then a totally different dynamics, called the Cahn-Hilliard equation. So, when looking for a result of convergence, we need to specify what the structure for the limiting flow should be (recall that the limiting flow is not taken in the same space, S uε ∈ Mε u ∈ N = Mε .) Another element that should come into play is possible time-rescalings as we pass to the limit ε → 0. 2.2 The result For simplicity we will reduce to the following case: Eε is family of C 1 functionals defined over M, an open subset of a Banach space B continuously embedded into a Hilbert space Xε (or of an affine space associated to a BaΓ nach). We assume Eε −→ F , with F a C 1 functional defined over N , open


277

set of a finite-dimensional vector space B embedded into a finite-dimensional Hilbert space Y . Definition 18 Eε Γ -converges along the trajectory uε (t) (t ∈ [0, T )) in the sense S to F if there exists u(t) ∈ N and a subsequence (still denoted uε ) S such that ∀t ∈ [0, T ), uε (t) u(t) and ∀t ∈ [0, T )

lim Eε (uε (t)) ≥ F (u(t)).

ε→0

Definition 19 If dEε (u), differential of Eε at u, is linear continuous on Xε , it is uniquely represented by a vector in Xε , denote it by ∇Xε Eε (u) (gradient for the structure Xε ), characterized by ∀φ ∈ Xε

d Eε (u + tφ) = dEε (u) · φ = ∇Xε Eε (u), φXε . dt |t=0

If this gradient does not exist, we use the convention ∇Xε Eε (u) Xε = +∞ For example, for the dynamics of Ginzburg-Landau, we wish to study the equation u ∂t u = ∆u + 2 (1 − |u|2 ). (12) |log ε| ε Let us see how to fit into the previous framework. Eε is defined on H 1 (Ω, C), we take B = H 1 (Ω, C) ⊂ L2 (Ω) and we define the Xε structure by 1 1 · 2L2 (Ω) , | · |2 = · 2Xε = |log ε| Ω |log ε| i.e. a rescaled version of L2 . Then B embeds continuously into Xε , and the gradient for the structure Xε is u ∇Xε Eε (u) = −|log ε| ∆u + 2 (1 − |u|2 ) . ε Indeed

u φ −∆u − 2 (1 − |u|2 ) ε Ω u 1 φ −| log ε| ∆u + 2 (1 − |u|2 ) = φ, ∇Xε Eε (u)Xε . = |log ε| Ω ε

dEε (u) · φ =

Then the PDE (12) is indeed exactly ∂t u = −∇Xε Eε (u) i.e. the gradientflow for the structure Xε . Recall that if a solution of ∂t uε = −∇Xε Eε (uε ), is smooth enough, we have

(13)

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Sylvia Serfaty

∂t uε , ∂t uε Xε = −∇Xε Eε (uε ), ∂t uε Xε = −∂t Eε (uε (t))

0

T

∂t uε 2Xε dt = Eε (uε (0)) − Eε (uε (T )).

Definition 20 A solution of the gradient-flow for Eε with respect to the structure Xε on [0, T ) is a map uε ∈ H 1 ([0, T ), Xε ) such that ∂t uε = −∇Xε Eε (uε ) ∈ Xε

for a.e. t ∈ [0, T ).

Such a solution is conservative if ∀t ∈ [0, T ) t ∂t uε (s) 2Xε ds Eε (uε (0)) − Eε (uε (t)) = 0

(this is true if uε is smooth enough). If uε is such a family of solutions on [0, T ) and Eε Γ -converges to F along uε (t) (in the sense of Definition 2), we define the energy-excess D(t) by Dε (t) = Eε (uε (t)) − F (u(t)) ≥ o(1) and D(t) = lim Dε (t) ≥ 0. ε→0

A family of solutions of the gradient flow is said to be well-prepared initially if D(0) = 0. Recall that F is always a lower bound for Eε . Also, it is always possible to have well-prepared initial data from assertion 2) (the construction part) of the Γ -convergence definition, Definition 1. We define similarly the gradient-flow for F for the structure Y . We can now state the abstract result. Theorem 5 ([19]) Let Eε and F be C 1 functionals over M and N respecΓ tively, Eε −→ F , and let uε be a family of conservative solutions of the flow of Eε ∂t uε = −∇Xε Eε (uε ) on [0, T ) (14) S

with uε (0) u0 , along which Eε Γ -converges to F in the sense of Definition 2. Assume moreover that 1) and either 2) or 2’) below are satisfied: S

1) (lower bound) For a subsequence such that uε (t) u(t), we have u ∈ H 1 ((0, T ), Y ) and ∀s ∈ [0, T ) s s ∂t uε 2Xε (t) dt ≥ ∂t u 2Y (t) dt. (15) lim ε→0

0

0

2) For any t ∈ [0, T ) lim ∇Xε Eε (uε (t) 2Xε ≥ ∇Y F (u(t)) 2Y .

ε→0

(16)


279

S

2 ) (construction) If uε u, for any V ∈ Y , any v defined in a neighborhood of 0 satisfying v(0) = u ∂t v(0) = V there exists vε (t) such that vε (0) = uε lim ∂t vε (0) 2Xε ≤ ∂t v(0) 2Y = V 2Y

ε→0

lim −

ε→0

d d Eε (vε ) ≥ − F (v) = −∇Y F (u), V Y dt |t=0 dt |t=0

Then if D(0) = 0 (i.e. the solution is well-prepared initially) we have D(t) = S 0 ∀t ∈ [0, T ), all inequalities above are equalities and ∀t ∈ [0, T ), uε (t) u(t) where ∂t u = −∇Y F (u) u(0) = u0 i.e. u is a solution of the gradient flow for F for the structure Y . 2.3 Interpretation This theorem means that under conditions 1) and 2), or 1) and 2’) (since 2’) implies 2)), solutions of the gradient flow of Eε for the structure Xε converge to solutions of limiting gradient-flow (for the structure Y ) if well-prepared. Let us make a few additional comments: 1. The limiting structure Y is somehow embedded in the conditions 1) and 2). The time rescalings are embedded in Xε . S S 2. In general we expect 1) and 2) to be satisfied for any uε u or uε (t) u(t) not necessarily solutions (here we required it only for solutions) 3. 1) and 2) do provide the extra C 1 order conditions on Γ -convergence. 2) in particular implies that critical points converge to critical points. 4. The difficulty is not in proving this theorem but in proving that in specific cases the conditions hold. 2.4 Idea of the proof Let us see how 1) and 2) imply the result. We assume ⎧ ⎪ ⎪ ∂t uε = −∇Xε Eε (uε ) ⎪ s s ⎪ ⎪ ⎪ ⎨ lim ∂ u 2 dt ≥ ∂ u 2 ε→0

0

t ε Xε

t

0

Y

⎪ ⎪ limε→0 ∇Xε Eε (uε ) 2Xε ≥ ∇Y F (u) 2Y ⎪ ⎪ ⎪ ⎪ ⎩ S uε (0) u0 , lim Eε (uε (0)) = F (u0 )

280

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Then, for all t < T we may write Eε (uε (0)) − Eε (uε (t)) = −

t

∇Xε Eε (uε (s)), ∂t uε (s)Xε ds 0

1 t ∇Xε Eε (uε ) 2Xε + ∂t uε 2Xε ds 2 0 1 t ∇Y F (u) 2Y | ∂t u 2Y ds − o(1) ≥ 2 0 t −∇Y F (u(s)), ∂t u(s)Y ds − o(1) ≥

=

(17)

0

= F (u(0)) − F (u(t)) − o(1) hence F (u(0)) − F (u(t)) ≤ Eε (uε (0)) − Eε (uε (t)) + o(1) But by well-preparedness Eε (uε (0)) = F (u(0)) + o(1) thus Eε (uε (t)) ≤ F (u(t)) + o(1). Γ

But, Eε −→ F implies limε→0 Eε (uε (t)) ≥ F (u(t)) therefore we must have equality everywhere and in particular equality in (17), that is t 1 t ∇Y F (u) 2Y + ∂t u 2Y ds = −∇Y F (u(s)), ∂t u(s)Y 2 0 0 or

t

∇Y F (u) + ∂t u 2Y ds = 0. 0

Hence, we conclude that ∂t u = −∇Y F (u), ∀t. The idea is thus to show that the energy decreases at least of the amount expected (i.e. the amount of decrease of F ), on the other hand it cannot decrease more because of the Γ -convergence, hence it decreases exactly of the amount expected, all along the trajectory. Proof of 2’) =⇒ 2) (2’) is a constructive proof of 2)). Observe that here uε does not depend on time. For every V ∈ Y we may pick v(t) such that v(0) = u ∂t v(0) = V i.e. pick a tangent curve to V at u. We assume there exists (we can construct) vε (t) such that ⎧ v (0) = uε ⎪ ⎪ ⎨ ε limε→0 ∂t vε (0) 2Xε ≤ V 2Y ⎪ ⎪ d d ⎩ lim ε→0 − dt |t=0 Eε (vε ) ≥ − dt |t=0 F (v) = −∇Y F (u), V Y


281

that is a curve vε (t) along which the energy decreases of at least the desired amount. Then, choosing V = −∇Y F (u), we have −∇Xε Eε (uε ), ∂t vε Xε = −

d Eε (vε ) ≥ −∇Y F (u), V Y = ∇Y F (u) 2Y dt |t=0

thus ∇Y F (u) 2Y ≤ −∇Xε Eε (uε ), ∂t vε (0))Xε ≤ ∇Xε Eε (uε ) Xε ∂t vε (0) Xε ≤ ∇Xε Eε (uε ) Xε ( V Y + o(1)) Recalling that V = −∇Y F (u), we conclude that ∇Xε Eε (uε ) Xε ≥ ∇Y F (u) Y + o(1). The idea was to rely on the fact that steepest descent is characterized as the evolution which maximizes the energy-decrease for a given ∂t uε 2 . We compare it to a test-evolution obtained by “pushing” uε in the direction V (and in fact choose the steepest descent direction V = −∇F (u)), i.e. find a curve v(t) and “lift it” to a curve vε that pushes uε in direction −∇Y F (u) with a decrease of energy of at least the expected one, and a cost ∂t vε 2 which is at most the expected one. We can in fact achieve this in such a way that ∂t uε (0) depends linearly on V . In “pedantic” terms, we show that there exists a linear embedding Iε : Tu N −→ Tuε M V → ∂t vε (0) which is an “almost-isometry” in the sense: lim Iε (V ) Xε = V Y

ε→0

and

lim Iε∗ ∇Xε Eε (uε ) = ∇Y F (u).

ε→0

2.5 Application to Ginzburg-Landau In order to retrieve the dynamical law for vortices, we need to prove that conditions 1) and 2’) of Theorem 5 can be proved for Ginzburg-Landau. As seen in Theorem 4, we need to consider the energies 1 1 |∇u|2 + 2 (1 − |u|2 )2 − πn| log ε| (18) Fε (u) = Eε (u) − πn|log ε| = 2 2ε Γ

and F = W (the renormalized energy) so that Fε −→ F . The structures we need are 1 · 2L2 (Ω) · 2Xε = | log ε| N = Ω n \diagonals (19) 1 · 2Y = · 2(R2 )n (20) π

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and a prescribed number of vortices of a priori fixed degrees ±1. Applying Theorem 5, we retrieve the dynamical law (first established by Lin and Jerrard-Soner) that the vortices flow according to a rescaled gradient-flow of the renormalized energy: Theorem 6 ([15, 12, 19]) Let uε be a family of solutions of u ∂t u = ∆u + 2 (1 − |u|2 ) |log ε| ε with either

uε = g on ∂Ω ∂u ε = 0 on ∂Ω ∂n

such that curl iuε , ∇uε (0) 2π

n

di δa0i

as ε → 0

i=1

with a0i distinct points in Ω, di = ±1, and Eε (uε )(0) − πn|log ε| ≤ Wd (a0i ) + o(1).

(21)

Then there exists T ∗ > 0 such that ∀t ∈ [0, T ∗ ), curl iu, ∇u(t) 2π

n

di δai (t)

i=1

as ε → 0, with

⎧ 1 dai ⎪ ⎨ = − ∇i Wd (a1 (t), · · · , an (t)) dt π ⎪ ⎩ ai (0) = a0i

(22)

where T ∗ is the minimum of the collision time and exit time (from Ω) of the vortices under this law. Moreover D(t) = 0 for every t < T ∗ . Thus, as expected, vortices move along the gradient flow for their interaction W , and this reduces the PDE to a finite dimensional evolution (a system of ODE’s). Thus result was obtained in [15, 12] (also in [23] for a certain regime with magnetic field), but with PDE methods, it is reproven in [19] with the Γ -convergence energetic method exposed here. 2.6 Remarks 1. The result holds as long as the number of vortices remains the initial one (so that the limiting configuration u = (a1 , · · · , an ) belongs to the same space N ). It ceases to apply when there are vortex-collisions or some vortex exits the domain under the law (22), even though these can happen. Then a further analysis is required, see Section 4.1 below.


283

2. Under the same hypotheses, if uε is a solution of the time-rescaled gradient S flow ∂t uε = −λε ∇Xε Eε (uε ) with D(0) = 0 then if λε 1, uε (t) u0 , ∀t S i.e. there is no motion; while if λε $ 1, uε (t) u, ∀t with ∇Y F (u) = 0 i.e. there is instantaneous motion to a critical point. Thus, we see that the structure Xε and the relation 1) in Theorem 5 contain the right timerescaling to see finite-time motion in the limit. For Ginzburg-Landau without magnetic field, it is necessary to accelerate the time by a |log ε| factor in order to see motion of the vortices (this is due to the fact that the renormalized energy W which drives the motion is a lower order term in the energy). 3. We can weaken conditions 1) and 2) to ε→0

t

∂t uε 2 ≥

lim 0

t

∂t u 2 − O(D(t)) 0

lim ∇Xε Eε (uε ) 2Xε ≥ ∇F (u) 2Y − O(D(t))

ε→0

where D(t) is the energy-excess, and handle the terms in D(t) in the proof via a Gronwall’s lemma (finally obtaining that D(t) ≡ 0 if D(0) = 0). 4. The method should and can be extended to infinite-dimensional limiting spaces and to the case where the Hilbert structures Xε and Y (in particular Y ) depend on the point, such as Yu = L2u , forming a sort of Hilbert manifold structure. It is thus interesting to see how, through Γ -convergence, the structures underlying the gradient-flows can become “curved” at the limit, even though they are not curved originally at the ε level, and also become possibly nonsmooth and nondifferentiable. In fact we can write down an analogue abstract result using the theory of “minimizing movements” of De Giorgi formalized by Ambrosio-Gigli-Savarè [1], a notion of gradient flows on structures which are not differentiable but simply metric structures. 5. The method works for Ginzburg-Landau with or without magnetic field as long as the number of vortices remains bounded. It is more difficult to apply it to other models such as Allen-Cahn, or 3D Ginzburg-Landau, because what misses is a more precise result and understanding on the profile of the defect during the dynamics. For example, for Allen-Cahn, we need to know that the energy-density remains proportional to the length of the underlying limiting curve during the dynamics (which is true a posteriori). It is also an open problem to apply it when the number of vortices is unbounded as ε → 0.

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3 Proof of the additional conditions for Ginzburg-Landau 3.1 A product-estimate for Ginzburg-Landau The relation 1) which relates the velocity of underlying vortices to ∂t uε can be read t 1 2 |∂t uε | ds ≥ π |dt ai |2 ds (23) lim ε→0 |log ε| [0,t]×Ω 0 i assuming curl iuε , ∇uε (t) 2π i di δai (t) , as ε → 0, ∀t. This turns out to hold as a general relation, without asking the configurations to solve any particular equation. It is related to the topological nature of the vortices. It can be embedded into the more general class of results of lower-bounds for Ginzburg-Landau functionals. The setting is now Ω a bounded domain of Rn (n ≥ 2) (we will need n = 3) and still Eε (u) = 12 Ω |∇u|2 + 2ε12 (1 − |u|2 )2 . We define the “current” ju associated to u as the 1-form ju = iu, du = k iu, ∂k udxk . Then the Jacobian Ju is the 2-form Ju =

1 1 d(ju) = diu, du 2 2

It can be identified to an (n − 2)-dimensional current through 1 Ju ∧ φ dx Ju(φ) = 2 for φ an (n−2)-form. This current corresponds to the vorticity lines in 3D. For example if u = eiφ with singular set γ straight line parallel to the z axis and a degree D around γ, then Ju = πD(dx ∧ dy) #γ i.e. given test vector-fields X and Y , X1 Y2 − X2 Y1 Ju(X, Y ) = D π 2 γ The total variation of Ju is then |Ju| = π|D|H1 (γ), multiple of the length of the line. Theorem 7 ([18]) Let uε be a family of H 1 (Ω, C) such that Eε (uε ) ≤ C|log ε| 0,β then up to extraction, ∀β > 0, Juε J in (CC (Ω))∗ , with Jπ an (n − 2)dimensional rectifiable integer-multiplicity current (see [13]) and for every X, Y continuous compactly supported vector fields, we have 6 1 |X · ∇uε |2 |Y · ∇uε |2 ≥ J(X, Y ) . (24) lim ε→0 |log ε| Ω Ω Ω


285

As a first corollary, in 2D, taking X = e1 , Y = e2 an orthonormal basis, we find 6 1 2 lim |∂1 uε | |∂2 uε |2 ≥ π |di | ε→0 |log ε| Ω Ω i 2 This estimate implies the estimate limε→0 |log1 ε| Ω |∇u2ε | ≥ π i |di | but is sharper. It implies in particular that if 12 Ω |∇uε |2 ≤ π i |di ||log ε| and di = ±1, then 1 |∇uε · X|2 = π |X(ai )|2 (25) ∀X, lim ε→0 |log ε| Ω i

i.e. there is isotropy in the repartition of the energy along different directions. In 3D, taking X ⊥ Y and maximizing the right-hand side of (24) over |X|, |Y | ≤ 1 we find 1 |∇uε |2 ≥ |J|(Ω), lim 2 ε→0 |log ε| Ω an estimate that was previously proved in [13]. Remark: Our result extends to higher energies Eε (uε ) ≤ Nε | log ε| with Nε unbounded, in that case we just need to rescale by Nε and replace J by the ε limit of Ju Nε . 3.2 Idea of the proof The method consists in reducing to two dimensions. By using partitions of unity, we can assume that X and Y are locally constant. We may then work in an open set U where X and Y are constant. If they are not parallel, they define a planar direction (if they are then J(X, Y ) = 0 and there is nothing to prove). We then slice U into planes parallel to that plane. Assume X = e1 and Y = e2 orthonormal vectors. In each plane we have the known 2D lower bounds of the type 1 |∇uε |2 ≥ π |di ||log ε| 2 plane∩U where di is the degree of the boundary of the balls, constructed with the ballconstruction method (see [17, 11, 20]). This is possible as long as there is a good bound on the energy on that planar slice, and the number of balls can be unbounded. The main trick is to observe that this is true for any metric in the plane, and use the metric λdx + λ1 dy, leading to λ 1 |∂1 uε |2 + |∂2 uε |2 ≥ π |di || log ε| 2λ plane∩U 2 plane∩U

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Integrating with respect to the slices yields λ 1 2 2 |∇uε · X| + |∇uε · Y | ≥ J(X, Y ) |log ε|. 2λ U 2 U U Optimizing with respect to λ, we conclude that 1 2 2 lim |∇uε · X| |∇uε · Y | ≥ J(X, Y ) ε→0 |log ε| U U U and we may finish by adding these estimates thanks to the partitions of unity. 3.3 Application to the dynamics In order to deduce a result for the dynamics in 2D, the idea is to use this theorem in dimension n = 3 with 2 coordinates corresponding to space coordinates and 1 coordinate corresponding to the time coordinate (this can be done in any dimension, but we restrict to 2D here for the sake of simplicity). The vortex-lines in 3D are then the trajectories in time of the vortex-points in 2D, and clearly the length of these lines is somehow related to the velocity of these points. Splitting the coordinates, we write = iu, ∂t u dt + iu, dspace u

ju Juε =

2

1 Vi dt ∧ dxi + dspace iu, dspace u 89 : 72 i=1 7 89 : Vε

µε →π

di δai (t)

Theorem 8 ([18]) Let uε (t, x) be defined over [0, T ] × Ω (Ω ⊂ Rn , here n = 2) and such that ⎧ ⎪ ⎨ ∀t Eε (uε (t)) ≤ C|log ε| ⎪ |∂t uε |2 ≤ C|log ε| ⎩ [0,T ]×Ω

then µε µ

0,γ ∗ in (CC ) ,

µ ∈ L∞ ([0, T ], C00 (Ω)∗ ),

µ=π

di δai (t)

i

Vε V,

V ∈ L∞ ([0, T ], C00 (Ω)∗ )

with dt µ + div V = 0. 0 0 ([0, T ] × Ω, Rn ), and f ∈ CC ([0, T ] × Ω), Moreover, ∀X ∈ CC 6 1 |X · ∇uε |2 f 2 |∂t uε |2 ≥ V · f X lim |log ε| ε→0 [0,T ]×Ω [0,T ]×Ω [0,T ]×Ω


287

In 2D, the vector V = (V1 , V2 ) really is π i di (∂t ai )δai (t) , such that ∂t π di δai (t) + div V = 0 i

Corollary 1 If in addition di = ±1 and ∀t, 12 Ω |∇uε |2 ≤ π ( i |di |) |log ε| (1 + o(1)), then for all intervals [t1 , t2 ) on which the ai ’s remain distinct, we have t2 1 2 |∂t uε | ≥ π |∂t ai |2 dt lim ε→0 |log ε| Ω×[t1 ,t2 ] t 1 i This is the desired estimate 1) in Theorem 5. To recall prove this corollary, that from (25), if Eε (uε ) πn|log ε| then |log1 ε| Ω |X · ∇uε |2 π i |X(ai )|2 and optimizing over X and f gives the L2 bound on V . 3.4 Proof of the construction 2 ) We wish to prove that 2’) holds for Ginzburg-Landau so that we deduce 2) S i.e. if uε u then limε→0 ∇Xε Eε (uε ) Xε ≥ ∇W (u) Y . Observe that this is a static result. We thus assume that curl iuε , ∇uε 2π i di δai , where u = ((a1 , d1 ), · · · , (an , dn )) and may consider disjoint balls B(ai , ρ) of fixed radius ρ. If ∇Xε E(uε ) Xε → +∞ there is nothing to prove. If ∇Xε Eε (uε ) Xε = O(1) then we can prove that Dε = o(1) where Dε = Eε (uε ) − πn|log ε| − W (u) is the “energy-excess”. The proof of this result (see [22, 21]) relies on the fact that ∇Xε Eε (u) Xε ≤ C means ∆u + u2 1 − |u|2 2 ≤ C = o(1) and one can take advantage of the ε |log ε| Ω fact that uε is thus an “almost-solution”. Once this is proved, we may deduce

1 2

1 2

|∇uε |2 = π|log ε| + O(1) B(ai ,ρ)

1 (1 − |uε |2 )2 ≤ Dε = o(1) 2ε2 ∇uε − iuε ∇⊥ Φ0 2 ≤ Dε = o(1)

|∇|uε ||2 + Ω\∪i B(ai ,ρ)

1 2

(26) (27)

Ω\∪i B(ai ,ρ)

where ∆Φ0 = 2π

di δai

in Ω

i

with the appropriate boundary conditions. The rough idea is that ∇ϕε ∇⊥ Φ0 outside of the vortex balls. Through these relations, everything is wellcontrolled outside the balls and inside the balls we shall only perform a pure translation.

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Given V = (V1 , . . . , Vn ), we want to push each ai in the direction Vi . For that purpose, define χt (x) = x + tVi in each Bi , and extend it in a smooth way outside of the Bi ’s into a family of smooth diffeomorphisms that keep ∂Ω fixed and are independent of ε. Choosing the deformation vε (x, t) = uε (χ−1 t (x)) does the job of pushing the vortices ai along the direction Vi . However it is not enough, and we need to add a phase correction ψt : vε (χt (u), t) = uε (x) eiψt (x)

(28)

so that for every t, the phase of vε is approximately the optimal one, that is the harmonic conjugate of di δai (t) ai (t) = ai + tVi . ∆Φt = 2π i

It is possible to construct ψt single-valued, independent of ε, so that ∇⊥ Φ0 + ∇ψt ∇⊥ (Φt ◦ χt ) . We will now check that the vε constructed this way works. First, 1 1 2 2 |∂t vε | (0) |Vi · ∇uε | + o(1) |log ε| Ω |log ε| i Bi π |Vi |2 + o(1) i

because χt achieves a translation of vector Vi in the Bi ’s while the contribution outside of the Bi ’s is negligible; and from the relation (25). The first requirement for 2’) is thus fulfilled. Let us check the second requirement, i.e. d E (vε (t)). With a change of the energy-decrease rate, by evaluating dt |t=0 ε variables, 2 1 1 2 |∇vε | + 2 1 − |vε |2 Eε (vε (t)) = 2 Ω 2ε −1 2 1 1 2 2 Dχt ∇(vε ◦ χt ) + 2 1 − |u| |Jac χt |. = 2 Ω 2ε Now, recall that χt is a translation in ∪i Bi hence |Jac χt | = cst there, while outside of ∪i Bi there is almost no energy, hence d d iψt 2 D χ−1 Eε (vε (t)) = |Jac χt | + o(1). t ∇ uε e dt |t=0 dt |t=0 Ω Next, we expand ∇ uε eiψt as ∇uε eiψt + iuε ∇ψt , expand the squares, and d . The crucial fact is that the terms which get differentiated do not apply dt |t=0 depend on ε. For the other terms, we use (27), so that there remains


d Eε (vε (t)) = dt |t=0

289

d ∇⊥ Φ0 · ∇⊥ Φ0 Dχ−1 t dt |t=0 Ω\∪i Bi d + ∇ψt · ∇⊥ Φ0 Ω\∪i Bi dt |t=0 1 d + |∇Φ0 |2 |Jac χt | + o(1) 2 dt |t=0 d 1 ⊥ 2 = |Dχ−1 t (∇ Φ0 + ∇ψt )| |Jac χt | + o(1) dt |t=0 2 Ω\∪i Bi

But observing that ψt was constructed in such a way that ∇⊥ Φ0 + ∇ψt = ∇⊥ (Φt ◦ χt ), and doing a change of variables again, we find d d 1 Eε (vε (t)) = |∇Φt |2 + o(1) dt |t=0 dt |t=0 2 Ω\∪i Bi d = Wd (a1 (t), · · · , an (t)) + o(1) dt |t=0 i.e. the desired result.

4 Extensions of the method 4.1 Collisions When there are some positive as well as some negative vortices, the limiting dynamics (22) induces collisions between vortices of opposite signs which attract each other. One expects that those vortices annihilate and the dynamics continues with the remaining ones. This has been treated recently in the papers [4, 5, 21]. One of the things done in [21] was to extend the method presented here to treat the case of collisions. If two vortices of opposite degrees get at a distance lε = o(1) of each other, then it is possible to rescale space in order to have two vortices at distance 1. As long as | log lε | |log ε| the same method we presented above carries through, i.e. we can prove the analogues of conditions 1) and 2), and yields that the dynamics continues with the same type of law (22) but in space-time rescaled coordinates. When vortices become too close to apply this, we focused on evaluating energy dissipation rates, through the study of the perturbed Ginzburg-Landau equation 1 (29) ∆u + 2 u(1 − |u|2 ) = fε in Ω, ε with Dirichlet or Neumann boundary data, where fε is given in L2 (Ω) (recall the instantaneous energy-dissipation rate in the dynamics is exactly |log ε| fε 2L2 (Ω) ). We prove that the energy-excess (still meaning the difference between Eε − πn|log ε| and the renormalized energy W of the underlying

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vortices) is essentially controlled by C fε 2L2 . We then show that when u solves (29) and has vortices which become forming what we call very close, an “unbalanced cluster” in the sense that i d2i = ( i di )2 in the cluster (see [21] for a precise definition), then the lower bound C C 2 , (30) fε L2 (Ω) ≥ min 2 l |log ε| l2 log2 l holds. In particular, when vortices get close to each other, say two vortices of opposite degrees for example, then they form an unbalanced cluster of vortices at scale l = their distance, and the relation (30) gives a large energydissipation rate (scaling like 1/l2 ). This serves to show that such a situation cannot persist for a long time and we are able to prove that the vortices collide and disappear in time Cl2 + o(1), with all energy-excess dissipating in that time. Thus after this time o(1), the configuration is again “well-prepared” and Theorem 6 can be applied again, yielding the dynamical law with the remaining vortices, until the next collision, etc... 4.2 Second order questions - stability issues We extended the “Γ -convergence” method to second order in order to treat stability questions for this 2D Ginzburg-Landau equation. Here is the abstract result, pushing the method of condition 2’) to second order. The setting is as in Section II.1. By stable critical point, we mean nonnegative Hessian. Theorem 9 ([22]) Let uε be a family of critical points of Eε with uε S u ∈ N , such that the following holds: for any V ∈ B , we can find vε (t) ∈ M defined in a neighborhood of t = 0, such that ∂t vε (0) depends on V in a linear and one-to-one manner, and vε (0) = uε (0) d dt |t=0 Eε (vε (t))

=

d dt |t=0 F (u

d2 dt2 |t=0 Eε (vε (t))

=

d2 dt2 |t=0 F (u

limε→0 limε→0

(31) + tV ) = dF (u).V + tV ) = Q(u)(V ).

(32) (33)

Then - if (31)-(32) are satisfied, then u is a critical point of F - if (31)-(32)-(33) are satisfied, then if uε are stable critical points of Eε , u is a stable critical point of F . More generally, denoting by n+ ε the dimension (possibly infinite) of the space spanned by eigenvectors of D2 Eε (uε ) associated to positive eigenvalues, and n+ the dimension of the space spanned by − eigenvectors of D2 F (u) associated to positive eigenvalues (resp. n− ε and n for negative eigenvalues); for ε small enough we have + n+ ε ≥n

− n− ε ≥n .


291

Thus, we reobtain that critical points converge to critical points of the limiting energy F (proved in [2] for Ginzburg-Landau), but in addition we obtain that under certain conditions, stability/instability of the critical point also passes to the limit. The previous result (Theorem 1), was an analysis of the C 1 structure of the energy landscape, thus suited to give convergence of gradientflow and critical points; while this is the C 2 analysis of the energy landscape around a critical point. For the Ginzburg-Landau energy, the construction done in Section 3.4 can be pushed to second order, yielding condition (33). Thus we deduce the corresponding theorem for solutions of (3), (see [22]). An interesting application is for Neumann boundary condition, for which it is known that the corresponding renormalized energy W has no stable critical point. Hence from Theorem 9 there can be no stable critical points of Eε with vortices (in contrast with the case of Gε with nonzero applied magnetic field). Theorem 10 ([22]) Let uε be a family of nonconstant solutions of −∆u = εu2 (1 − |u|2 ) in Ω ∂u on ∂Ω ∂n = 0 (on Ω ⊂ R2 simply connected) such that Eε (uε ) ≤ C|log ε|; then, for ε small enough, uε is unstable. This is an extension of a result of Jimbo and Sternberg [14] for convex domains.

References 1. Ambrosio, L.; Gigli, N.; Savarè, G. Gradient flows in metric spaces and in the Wasserstein spaces of probability measures, Birkha¨ user, 2005. 2. Bethuel, F.; Brezis, H.; Hélein, F. Ginzburg-Landau vortices. Progress in Nonlinear Partial Differential Equations and Their Applications, 13. Birkha¨ user Boston, Boston, 1994. 3. Bethuel, F.; Brezis, H.; Orlandi, G. Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions. J. Funct. Anal. 186 (2001), no. 2, 432–520. 4. Bethuel, F.; Orlandi, G.; Smets, D. Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics. Duke Math. J., to appear. 5. Bethuel, F.; Orlandi, G.; Smets, D. Quantization and motion laws for GinzburgLandau vortices. Preprint, 2005. 6. Chen, X. Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Eq. 96 (1992), 116-141. 7. DeGennes, P. G. Superconductivity of metal and alloys. Benjamin, New York and Amsterdam, 1966. 8. De Mottoni, P; Schatzman, M. Development of interfaces in RN , Proc. Roy. Soc. Edinburgh, Sec A 116 (1990), 207-220. 9. Evans, L.C; Soner, H.M; Souganidis, P.E. Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45, No 9, (1992), 1097– 1123.

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10. Ilmanen, T. Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Diff. Geom. 38 (1993), 417-461. 11. Jerrard, R. L. Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30 (1999), no. 4, 721-746. 12. Jerrard, R. L.; Soner, H. M. Dynamics of Ginzburg-Landau vortices. Arch. Ration. Mech. Anal. 142 (1998), no. 2, 99–125. 13. Jerrard, R. L.; Soner, H. M. The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations 14 (2002), no. 2, 151–191. 14. Jimbo, S.; Sternberg, P. Nonexistence of permanent currents in convex planar samples. SIAM J. Math. Anal. 33 (2002), no. 6, 1379–1392. 15. Lin, F.-H. Some dynamic properties of Ginzburg-Landau vortices. Comm. Pure Appl. Math 49 (1996), 323–359. 16. Lin, F.-H.; Rivière, T. A quantization property for static Ginzburg-Landau vortices. Comm. Pure Appl. Math. 54 (2001), no. 2, 206–228. 17. Sandier, E. Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152 (1998), no. 2, 379–403; Erratum, Ibid, 171 (2000), no. 1, 233. 18. Sandier, E.; Serfaty, S. A product-estimate for Ginzburg-Landau and corollaries. J. Funct. Anal. 211 (2004), no. 1, 219–244. 19. Sandier, E.; Serfaty, S. Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math. 57 (2004), no. 12, 1627–1672. 20. Sandier, E.; Serfaty, S. Vortices in the Magnetic Ginzburg-Landau Model, Birkha¨ user, to appear. 21. Serfaty, S. Vortex-collision and energy dissipation rates in the Ginzburg-Landau heat flow, part I: Study of the perturbed Ginzburg-Landau equation; part II: The dynamics. Preprint, 2005. 22. Serfaty, S. Stability in 2D Ginzburg-Landau passes to the limit. Indiana Univ. Math. J 54 (2005), no. 1, 199–222. 23. Spirn, D. Vortex motion law for the Schrdinger-Ginzburg-Landau equations. SIAM J. Math. Anal. 34 (2003), no. 6, 1435–1476. 24. Tinkham, M. Introduction to superconductivity. Second edition. McGraw-Hill, New York, 1996.

The author wishes to thank warmly the organizers of the mini-course, and in particular Andrea Braides, for their invitation and hospitality.

PDE analysis of concentrating energies for the Ginzburg-Landau equation Didier Smets Laboratoire Jacques-Louis Lions, Université de Paris 6 4 place Jussieu, Boite Courier 187, 75252 Paris, France e-mail: [email protected]

These notes are intended as a complement to the lectures given by the author during the summer course Concentration phenomena in variational problems which took place in Rome, Universit` a La Sapienza, September 1 - 5 2003. The author wishes to thank Andrea Braides and Valeria Chiad` o Piat for their kind invitation and constant help. The great majority of the material will be taken from [3, 4, 5], but we tried to adopt a more explanatory viewpoint. Some new material is also presented, mainly in Section 4. By no mean we wish to discuss the whole history of the analysis of the Ginzburg-Landau equations here; for this purpose the interested reader is highly encouraged to consult the references in [2, 3, 5]. There are actually many equations which are called the Ginzburg-Landau equations; some have a physical motivation, others don’t. Nevertheless, they all have some common mathematical features, and for this reason we will restrict our analysis in these notes to the simplest mathematical case: the static Ginzburg-Landau equation without magnetic field (which is also the less physical!), but in dimension N ≥ 3. The brief discussion of the time dependent (parabolic and Schr¨ odinger) equations, which was the object of the last lecture, is not covered in these notes.

1 Introduction Let Ω be a bounded smooth domain in RN , N ≥ 3, and 0 < ε < 1 be given. A function uε : Ω → C is said to satisfy the Ginzburg-Landau equation (GL)ε on Ω if (GL)ε

−∆uε =

1 uε (1 − |uε |2 ) ε2

on Ω.

We do not prescribe any boundary condition here since we will only discuss results which are local in nature.

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Didier Smets

To the Ginzburg-Landau equation is associated an energy density eε (uε ) =

(1 − |uε |2 )2 |∇uε |2 + , 2 4ε2

and a total energy

Eε (uε ) =

eε (uε ). Ω

In the sequel, we assume that we are given a sequence ε → 0 and a corresponding sequence of solutions uε of (GL)ε on Ω. The main assumption that we will make in some places (see [2] for the motivation) is (H0 )

Eε (uε ) ≤ M0 |log ε|,

where M0 is some fixed constant not depending on ε. Under this assumption, the renormalized energy densities µε (uε ) =

eε (uε ) |log ε|

are uniformly bounded in L1 (Ω), and we may assume, passing possibly to a subsequence, that as measures µε (uε ) → µ∗ for some non-negative Radon measure µ∗ on Ω. The measure µ∗ is called the energy defect measure or energy concentration measure, and the primary goal of these notes is to present some PDE tools for (GL)ε that lead to qualitative and quantitative properties of µ∗ . In Section 2, after discussing briefly the scaling properties of (GL)ε , we introduce the monotonicity formula: a simple tool which already allows to obtain important information concerning µ∗ . Section 3 is devoted to the clearing-out theorem and to its numerous consequences. The proof of this theorem is too long to be covered in a lecture, and is therefore omitted. Instead, we will concentrate on its consequences concerning µ∗ . In section 4, we present some (sometimes new) material concerning pointwise estimates, potential estimates, ..., which will enter into play in Section 5. In this last section, we gather all the information already obtained concerning µ∗ , and derive some of its regularity properties as well as a curvature equation.

2 The monotonicity formula 2.1 Scaling properties One of the most important properties of (GL)ε is its scaling property. Let 0 < R < 1ε and " = εR. If uε is a solution of (GL)ε on Ω and x ∈ Ω, then the rescaled function

PDE analysis of concentrating energies for the Ginzburg-Landau equation

295

y−x ) R is a solution of (GL) on ΩR = R(Ω − x). Notice also that v (y) = uε (

E (v , ΩR ) = RN −2 Eε (uε , Ω).

(1)

The following definition is motivated by this last equality. Definition 21 Let x ∈ Ω and r > 0 such that B(x, r) ⊂ Ω. The rescaled energy of uε in B(x, r) is defined by 1 ˜ eε (uε ). Eε (uε , x, r) = N −2 r B(x,r) Coming back to (1), we therefore obtain the identity ˜ε (uε , x, r). ˜ (v , x, rR) = E E

(2)

2.2 The monotonicity formula We first start with the following identity: Lemma 1 (Pohozaev identity) If uε is a solution of (GL)ε on B(x, r), then N N −2 2 |∇uε | + 2 (1 − |uε |2 )2 2 4ε B(x,r) B(x,r) (3) 2 r ∂uε |∇ uε |2 r 2 2 − = [r + (1 − |u | ) ]. ε 2 2 ∂n 4ε2 ∂B(x,r) Proof. Integrating by parts, we obtain −∆uε (y)(y − x) · ∇uε (y) dy = B(x,r)

∇uε · ∇ ((y − x) · ∇uε )

B(x,r)

−

(4) r|∂r uε |2 .

∂B(x,r)

Notice that ∇uε · ∇ ((y − x) · ∇uε ) =

1 ∇ |∇uε |2 · (y − x) + |∇uε |2 . 2

Therefore, integrating by parts once more leads to N −2 −∆uε (y)(y − x) · ∇uε (y) dy = − |∇uε |2 2 B(x,r) B(x,r) 1 2 + r|∇uε | − r|∂r uε |2 . 2 ∂B(x,r) ∂B(x,r)

(5)

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Didier Smets

On the other hand, 1 (1 − |uε |2 )2 2 · (y − x) u (1 − |uε | )(y − x) · ∇uε = −∇ 2 ε 4ε2 B(x,r) ε B(x,r) (1 − |uε |2 )2 (1 − |uε |2 )2 − r . (6) =N 4ε2 4ε2 B(x,r) ∂B(x,r) Since uε satisfies (GL)ε , we obtain the desired result combining (5) and (6). We are now in position to prove the following proposition, which possesses many consequences, as we will see. Proposition 22 (Monotonicity formula) Assume uε is a solution of (GL)ε in B(x, R), then ∂uε 2 1 d ˜ (1 − |uε |2 )2 + 1 (Eε (uε , x, r)) = N −2 dr r rN −1 B(x,r) 2ε2 ∂B(x,r) ∂n for 0 < r < R. Proof. First one has, d (Eε (uε , x, r)) = dr

∂B(x,r)

1 |∇uε |2 + 2 2 4ε

∂B(x,r)

1 1 |∇ uε |2 + + 2 (1 − |uε |2 )2 . 2 2 ∂n 4ε

=

(1 − |uε |2 )2 ∂B(x,r) ∂uε 2

Hence, d ˜ (Eε (uε , x, r)) dr

2 1 ∂uε (1 − |uε |2 )2 |∇ uε |2 + + = − N −1 Eε (uε , x, r) + N −2 r r 2 ∂n 4ε2 ∂B(x,r) 2 N −2 N −2 |∇uε |2 + 2 N −1 =− (1 − |uε |2 )2 rN −1 B(x,r) 2 4ε r B(x,r) 2 1 ∂uε 1 1 |∇ uε |2 + + 2 (1 − |uε |2 )2 + N −2 r 2 2 ∂n 4ε ∂B(x,r) N N −2 1 2 2 2 |∇uε | + 2 =− [ (1 − |uε | ) ] rN −1 B(x,r) 2 4ε B(x,r) 1 (1 − |uε |2 )2 + 2 N −1 2ε r B(x,r) 2 1 ∂uε 1 1 |∇ uε |2 2 2 + + N −2 + 4ε2 (1 − |uε | ) . r 2 2 ∂n ∂B(x,r) N −2

1


297

Using Lemma 1, we obtain 3 2 2 1 ∂uε 1 |∇ uε |2 d ˜ 1 2 2 (Eε (uε , x, r)) = − N −2 − + 2 (1 − |uε | ) dr r 2 2 ∂n 4ε ∂B(x,r) 2 3 2 1 1 ∂uε 1 |∇ uε |2 2 2 + N −2 + + 2 (1 − |uε | ) r 2 2 ∂n 4ε ∂B(x,r) 2 ∂uε 1 (1 − |uε |2 )2 + 1 , = N −2 N −1 r r 2ε2 ∂B(x,r) ∂n B(x,r) which yields the result. 2.3 Consequences for µ∗ In this section, we will already derive two important consequences of the monotonicity formula. The first one can be directly translated into a property of the defect measure µ∗ . Let us first recall or set some general definitions about densities of measures. Definition 23 Let ν be a non-negative Radon measure in RN . For m ∈ N, the m-dimensional lower density of ν at the point x is defined by Θ∗,m (ν, x) = lim inf r→0

ν(B(x, r)) , ωm r m

where ωm denotes the volume of the unit ball B m . Similarly, the m-dimensional ∗ (ν t , x) is given by upper density Θm ∗ (ν, x) = lim sup Θm r→0

ν(B(x, r)) . ωm r m

When both quantities coincide, ν admits a m-dimensional density Θm (ν, x) at the point x, defined as the common value. Proposition 24 Let K ⊂ Ω be any compact set and assume that (H0 ) holds. Then, there exists a constant C(K) depending only on K such that ∗ ΘN −2 (µ∗ , x) ≤ C(K)M0

for all x ∈ K.

(7)

Proof. Let x ∈ K and R = dist(x, ∂Ω). For r < R, we have by the monotonicity formula: 1 1 e (u ) ≤ eε (uε ) ≤ dist(K, ∂Ω)2−N M0 |log ε|. ε ε rN −2 B(x,r) RN −2 B(x,R) Dividing by |log ε| and letting ε go to zero we obtain dist(K, ∂Ω)2−N µ∗ (B(x, r)) ≤ M0 . ωN −2 rN −2 ωN −2 The conclusion follows taking the limsup as r goes to zero.

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Remark 1. i) Notice that the constant appearing in the previous proposition diverges as K approaches the boundary of Ω. This is unavoidable since we have not imposed any condition on the boundary of Ω. ii) It also immediately follows from the monotonicity formula that the lower and upper N −2 densities coincide, i.e. that µ∗ has a N − 2 dimensional density everywhere in Ω. We will come back to this later, when dealing with regularity properties. Comment. Geometrically speaking, the bounds on the N − 2 dimensional density of µ∗ shows that this measure cannot be concentrated on objects of lower dimension than N − 2. For example, for N = 3 the measure µ∗ cannot charge a point (as it is the case in dimension 2 [2]). There is still a wide variety of objects ranging between the dimensions N − 2 and N ; our subsequent analysis will restrict the possible situations much more! The next consequence of the monotonicity formula, which we will derive now, deals with estimates on the potential part of the energy. It is a new estimate, which we will use in a crucial way in Section 4. For later convenience, we now fix the notation for the potential and define Vε (uε ) =

(1 − |uε |2 )2 . 4ε2

Proposition 25 Assume that uε solves (GL)ε on Ω. Let x ∈ Ω and 0 < R0 < R1 such that B(x, R1 ) ⊂ Ω. There exists r(x) ∈ (R0 , R1 ) such that ˜ε (uε , x, R1 ) E 1 log Vε (uε ) ≤ eε (uε ). (8) ˜ε (uε , x, R0 ) log(R1 /R0 ) E B(x,r(x)) B(x,r(x)) ˜ε (uε , x, r) and g(r) = r2−N Proof. Define f (r) = E from the monotonicity formula that 1 d f (r) ≥ g(r) dr r

B(x,r)

Vε (uε ). It follows

for all r ∈ (R0 , R1 ).

(9)

Let F and G be defined by the change of variable f (r) = F (log(r)) and g(r) = G(log(r)). Inequality (9) is transported into the inequality d F (s) ≥ G(s) ds

for all s ∈ (log(R0 ), log(R1 )).

It suffices then to infer the next ODE lemma and the conclusion follows. Lemma 2 Let F and G be absolutely continuous non-negative functions on the interval [a, b] ⊂ R. Assume that F ≥ G a.e. on [a, b], then there exists c ∈ (a, b) such that F (b) 1 log F (c). G(c) ≤ b−a F (a)


Proof. Let λ =

1 b−a

log

F (b) F (a)

299

. If G(s) > λF (s) everywhere in (a, b), and

since by assumption F ≥ G, we obtain F > λF so that d (exp(−λs)F (s)) > 0 ds

on (a, b).

Integrating the previous inequality on (a, b) we are led to F (b) > exp(−λ(a − b))F (a) = F (b), a contradiction. Comment. A particularly useful choice of R0 and R1 in Proposition 25 is given by R0 = εα and R1 = εβ for some 0 ≤ β < α < 1. Indeed, for this ˜ε (uε , x, εα ) and choice we have log(R1 /R0 ) |log ε|, and assuming that E β ˜ Eε (uε , x, ε ) are of the same order we deduce from (8) that the potential energy on B(x, r(x)) is |log ε| times smaller than the total energy on the same ball.

3 Regularity theorems for (GL)ε 3.1 Bochner, small energy regularity and clearing-out A major difficulty in proving regularity theorems for (GL)ε stems from the factor ε−2 which appears in the equation. A key ingredient in this viewpoint is given by the following Bochner identity: Proposition 26 (Bochner’s identity) If uε satisfies (GL)ε , then the energy density eε ≡ eε (uε ) verifies −∆eε = −|D2 uε | − |∆uε |2 +

1 − |uε |2 2 |∇uε |2 − 2 |uε |2 |∇|uε ||2 , 2 ε ε

so that in particular −∆eε ≤ Ce2ε .

(10)

Proof. We have, since uε verifies (GL)ε , |∇uε |2 = −∆(∇uε ) · ∇uε − |D2 uε |2 = −∇(Vε (uε )) · ∇uε − |D2 uε |2 −∆ 2 and −∆ (Vε (uε )) = −∇(Vε (uε )) · ∇uε − Vε (uε )∆uε = −∇(Vε (uε )) · ∇uε − |∆uε |2 . Bochner’s identity follows by addition.

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Note that (10) is not far from saying that eε is subharmonic, in which case regularity estimates would follow very easily. The following theorem is an important consequence of the Bochner identity, it was first proved by R. Schoen for harmonic maps (see [8, 6] for proofs in a very similar context). Theorem 27 (Small energy regularity) Let v be a solution of (GL)ε on the ball B(R) for some √ R > 0. There exists a constant γ0 > 0, depending only on N such that if R > ε and if 1 e (v ) ≤ γ0 , RN −2 B(R) then e (v )(x) ≤

sup x∈B(R/2)

C RN

e (v )

(11)

B(R)

where the constant C depends only on N . In general, an estimate like (11) does not hold everywhere for solutions of (GL)ε , in particular not in the so called ”vorticity set” where |uε | ≤ 1/2. The following theorem will therefore play a central role (see [3] for the proof, which uses the monotonicity formula in a crucial way): Theorem 28 (Clearing-out) Let 0 < σ ≤ 12 be given. There exist a constant η0 > 0√depending only on σ and N such that if uε verifies (GL)ε on B(R), R > ε and if 1 eε (uε ) ≤ η0 |log ε|, RN −2 B(R) then |uε | ≥ 1 − σ

on B(

R ). 2

In the next proposition, we prove that away from the vorticity set an estimate like (11) holds. Proposition 29 Let uε be a solution of (GL)ε on B(R) such that uε verifies assumption √ (H0 ). There exist a constant 0 < σ ≤ 12 depending only on N such that if R > ε and if |uε | ≥ 1 − σ then sup x∈B(R/2)

eε (uε )(x) ≤

on B(R), C RN

eε (uε ), B(R)

where the constant C depends only on M0 and N .


301

Proof. Using the scaling properties of (GL)ε , we may assume without loss of generality that R = 1. Since by assumption |uε | ≥ 1 − σ ≥ 12 on B(1), there is some real-value function ϕε defined on B(1) such that uε = ρε exp(iϕε )

in B(1),

(12)

where ρε = |uε |. Changing uε possibly by a constant phase, we may impose the additional condition 1 ϕε = 0. (13) |B(1)| B(1) We split as previously the estimates for the phase ϕε and for the modulus ρε , and we begin with the phase. Inserting (12) into (GL)ε we are led to the elliptic equation in B(1). (14) −div(ρε ∇ϕε ) = 0 In contrast with the equation for the modulus which we will tackle later, (14) has the advantage that the explicit dependence on ε has been removed. We will handle (14) as a linear equation for the function ϕε , ρε being considered as a coefficient. In the sequel, we write ϕ = ϕε and ρ = ρε when this is not misleading. In order to avoid boundary conditions, we consider the truncated function ϕ˜ defined by ϕ˜ = ϕ χ, where χ is a smooth cut-off function such that 4 χ ≡ 1 on B( ) 5

and

5 χ ≡ 0 on RN \ B( ). 6

The function ϕ˜ then verifies the equation −div(ρ∇ϕ) ˜ = div(ρ2 ϕ∇χ) + ρ2 ∇χ · ∇ϕ

in B(1).

(15)

Moreover, by construction 4 supp(ϕ) ˜ ⊂ B( ). 5 Since by assumption ρ is close to 1, it is natural to treat the l.h.s. of (15) as a perturbation of the Laplace operator, and to rewrite (15) as follows −∆ϕ˜ = div((ρ2 − 1)∇ϕ) ˜ + div(ρ2 ϕ∇χ) + ρ2 ∇χ · ∇ϕ

in B(1).

We introduce the function ϕ0 defined on B(1) as the solution of −∆ϕ0 = div(ρ2 ϕ∇χ) + ρ2 ∇χ · ∇ϕ in B(1), on ∂B(1). ϕ0 = 0

(16)

In particular, since χ ≡ 1 on B( 45 ), we have −∆ϕ0 = 0

4 in B( ). 5

(17)

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Didier Smets

We set ϕ1 = ϕ˜ − ϕ0 , i.e. ϕ˜ = ϕ0 + ϕ1 . We will show that ϕ1 is essentially a perturbation term. At this stage, we divide the estimates into several steps. We start with linear estimates for ϕ0 . Step 1: Estimates for ϕ0 . We claim that 2 ∇ϕ0 2L2∗ (B(1))

3

≤ C1

eε (uε ) 2

and ∇ϕ0 2L∞ (B(3/4)) where 2∗ =

2N N −2

(18)

B(1)

3

≤ C2

eε (uε ) ,

(19)

B(1)

is the Sobolev exponent in dimension N .

Proof. The first estimate follows from the linear theory for the Laplace operator, whereas the second follows from the first one and the fact that ϕ0 is harmonic on B(4/5). Step 2: The equation for ϕ1 . The function ϕ1 verifies the elliptic problem −∆ϕ1 = div((ρ2 − 1)∇ϕ) ˜ in B(1), (20) on ∂B(1). ϕ1 = 0 It is convenient to rewrite equation (20) as −∆ϕ1 = div((ρ2 − 1)∇ϕ1 ) + div (g0 ),

(21)

where we have set g0 = (ρ2 −1)∇ϕ0 . Using (18), we obtain, for any 2 ≤ q < 2∗ , the estimate for g0 g0 qLq (B(1)) ≤ C (M0 |log ε|)

2∗ −q 2∗

ε

2(2∗ −q) 2∗

q

eε (uε ) L2 1 .

(22)

Indeed,

|ρ2 − 1|q |∇ϕ0 |q ≤ B(1)

|∇ϕ0 |2

∗

2q∗

B(1)

|ρ2 − 1|

2∗ q 2∗ −q

2∗2−q ∗

B(1) q 2

≤ C eε (uε ) L1 ε

2(2∗ −q) 2∗

B(1)

|ρ2 − 1|2 4ε2

2∗2−q ∗

and the conclusion follows. We now estimate ϕ1 from (21) through a fixed point argument.

,


303

Step 3: The fixed point argument. Equation (21) may be rewritten as ϕ1 = T (div((ρ2 − 1)∇ϕ1 )) + T (div g0 ), which is of the form (Id − A)ϕ1 = b −1

where T = ∆ , A is the linear operator v → T (div((ρ2 − 1)∇v)) and b = T (div g0 ). Consider the Banach space Xq = W01,q (B(1)). It follows from the linear theory for T that A : Xq → Xq is linear continuous and that A L(Xq ) ≤ C(q) 1 − ρ L∞ (B(1)) . In particular, we may fix σ > 0 such that C(q) 1 − ρ L∞ (B(1)) ≤ C(q)σ
2, the previous estimate presents a substantial improvement over the corresponding inequality with q replaced by 2, which follows directly from (H0 ). This improvement is crucial in order to prove the smallness of both the modulus and potential terms in the energy, which we derive now. Step 5: Estimates for the modulus and potential terms. The function ρ satisfies the equation −∆ρ + ρ|∇ϕ|2 = ρ

(1 − ρ2 ) . ε2

(26)

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Didier Smets

Since χ ≡ 1 on B( 45 ), we have ϕ = ϕ˜ on B( 45 ). Let ξ be a non-negative cut-off function such that ξ ≡ 1 on B( 34 ) and ξ ≡ 0 outside B( 45 ). Multiplying (26) by (1 − ρ2 )ξ and integrating by parts we obtain (1 − ρ2 )2 2 2 2ρ|∇ρ| ξ+ ρ = ∇ρ·∇ξ(1−ρ )+ ρ(1−ρ2 )|∇ϕ| ˜ 2 ξ. ε2 B(1) B(1) B(1) B(1) Hence, since ρ ≥

1 2

on B(1) we obtain 12

|∇ρ|2 + Vε (uε ) ≤ Cε B(3/4)

12

|∇ρ|2

Vε (uε )

B(1)

B(1)

q2

|∇ϕ| ˜

q−2 q

q

+C B( 45 )

(1 − ρ ) 2

B( 45 )

so that using (25) we finally infer that 2 [ ∇|uε | + Vε (uε )] ≤ C(M0 )εα B(3/4)

eε (uε ),

q q−2

(27)

B(1)

for some α > 0 [here we have fixed 2 < q < 2∗ ]. To summarize, we have proved at this stage that eε (uε ) ≤ |∇ϕ0 |2 + rε , for some rε ≥ 0 which verifies rε ≤ C(M0 )εα B(3/4)

eε (uε ),

(28)

(29)

B(1)

for some small α > 0 depending only on N . Therefore, we set Step 6: Proof of Proposition 29 completed. We are √ going to apply Theorem 27 to a suitably scaled version of uε . Let ε < r0 < 18 , to be determined later, set " = rε0 and let x0 ∈ B(1/2) be fixed. Consider the map v defined on B(1) by x − x0 v (x) = uε r0 so that uε (x0 ) = v (0). By scaling, we have 1 e (v ) = N −2 eε (uε ). r0 B(1) B(x0 ,r0 )

(30)

Note in particular, since r0 < 18 , that B(x0 , r0 ) ⊂ B(3/4), and we may apply the decomposition (28) to assert that


eε (uε ) ≤ meas(B(x0 , r0 ) ) · ∇ϕ0 2L∞ +

305

rε

B(x0 ,r0 )

B(3/4)

≤ Cr0N eε (uε ) L1 (B(1)) + C(M0 )εα eε (uε ) L1 (B(1)) . Hence, going back to (30) e (v ) ≤ Cr02 eε (uε ) L1 (B(1)) + C(M0 )r02−N εα eε (uε ) L1 (B(1)) .

(31)

B(1)

Therefore, we choose r0 = inf

1 2C eε (uε ) L1 (B(1)) − 1 ,( ) 2 8 γ0

.

Note in particular that r02−N diverges at most as |log ε| sufficiently small, C(M0 )r02−N εα eε (uε ) L1 (B(1)) ≤

N −2 2

. Hence, for ε

γ0 . 2

On the other hand, by construction, ωN C2 (Λ)r02 eε (uε ) L1 (Λ) ≤

γ0 . 2

Applying Theorem 27 to v for R = 1, we therefore deduce that e (v ) r02 eε (uε )(x0 ) = e (v )(0) ≤ C B(1)

≤ C(M0 )r02 eε (uε ) L1 (B(1)) + C(M0 )r02−N εα eε (uε ) L1 (B(1)) , so that for ε sufficiently small eε (uε )(x0 ) ≤ C(M0 )

eε (uε ), B(1)

and the proof is complete. Combining Theorem 27 with Proposition 29 we derive Theorem 30 Assume that uε is a solution of (GL)ε on B(x, R) and that (H0 ) holds. √ There exist a constant η1 depending only on N and M0 such that if R > ε and if 1 eε (uε ) ≤ η1 |log ε| RN −2 B(R) then eε (uε ) ≤

C RN

eε (uε ) B(R)

where C depends only on N and M0 .

on B(

R ), 4

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Didier Smets

3.2 Consequences for µ∗ We are now in position to obtain much more information on the concentration measure µ∗ . Recall that we already know that the N − 2 dimensional density of µ∗ is locally bounded. Proposition 31 There exists η > 0 such that for every x ∈ Ω, either

ΘN −2 (µ∗ , x) ≥ η

or

ΘN −2 (µ∗ , x) = 0.

Moreover, if

ΘN −2 (µ∗ , x) = 0

then

∗ ΘN (µ∗ , x) < +∞.

Proof. Define η = ωηN1 . Assume that x ∈ Ω is such that ΘN −2 (µ∗ , x) < η. By definition, there exists R > 0 (which we can assume small enough so that B(x, R) ⊂ Ω) such that µ∗ (B(x, R)) < η. ωN RN −2 Hence, for every ε sufficiently small 1 eε (uε ) < η1 |log ε|. RN −2 B(x,R) Applying Theorem 30 we obtain 1 eε (uε ) < C(M0 )R−2 η1 |log ε| eε (uε ) ≤ C(M0 ) N R B(x,R) on B(x, R/4). In particular, for any r < R/4 µ∗ (B(x, r)) ≤ rN R−2 η1 and the conclusion follows taking the limsup as r goes to zero. In view of the previous proposition, we introduce the concentration set Σµ := {x ∈ Ω s.t. ΘN −2 (µ∗ , x) > 0} . Proposition 32 The concentration set Σµ is closed in Ω and of locally finite HN −2 Hausdorff measure. Proof. Since by the previous proposition Σµ := {x ∈ Ω s.t. ΘN −2 (µ∗ , x) ≥ η} , its closedness follows from the upper semi-continuity of the density. Now, let K ⊂ Ω be any compact subset and set δ0 = dist(K, ∂Ω). Fix δ < δ0 /2 and consider a standard (say parallepipedic) covering of RN such that


RN ⊆ ∪j∈I B(xj , δ),

and

307

δ δ B(xi , ) ∩ B(xj , ) = ∅ for i = j. 2 2

Set Iδ = {i s.t. B(xi , δ) ∩ Σµ = ∅} . For i ∈ Iδ , there exists some yi ∈ Σµ ∩ B(xi , δ). Hence, by Proposition 31, µ∗ (B(yi , δ) > ηωN δ 2−N , and in particular µ∗ (B(xi , (2δ)) > ηωN δ 2−N .

(32)

B(xi , 2δ )

are disjoints, the balls B(xi , 2δ) On the other hand, since the balls cover at most C times RN , where C is a constant depending only on N . Therefore, µ∗ (B(xi , 2δ)) ≤ CM0 . (33) i∈Iδ

Combining (32) and (33) we obtain $Iδ ≤ CM0 δ 2−N . Since by definition, HN −2 (Σµ ) ≤ C lim supδ→0 ($Iδ )δ N −2 , the conclusion follows. We may now state and prove our main structure and regularity theorem concerning the measure µ∗ and its concentration set Σµ . We first recall some the notion of rectifiability. Definition 33 A set Σ ⊂ RN is said to be m-rectifiable (0 ≤ m ≤ N ) if Hm almost all of Σ can be covered by the union of countably many Lipschitz images of B m . A Radon measure ν on RN is said to be m-rectifiable if there exist an m-rectifiable set Σ and a nonnegative density function h ∈ L1loc (RN , Hm Σ) such that ν = h(·) Hm Σ. Theorem 34 (Structure and regularity) The set Σµ is (N −2)-rectifiable and the measure µ∗ can be exactly decomposed as µ∗ = g(.)HN + Θ∗ (.)HN −2

Σµ ,

∞ N −2 where g ∈ L∞ loc (Ω) and the density function Θ∗ ∈ Lloc (Ω, H

Σµ ).

Proof. The rectifiability of Σµ follows from the lower bound on the density given in Proposition 31 and Preiss regularity theorem [7]. Since Σµ is closed, µ∗ = µ∗

(Ω \ Σµ ) + µ∗

Σµ .

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Didier Smets

By propositions 31 and 32, µ∗ Σµ is absolutely continuous with respect to HN −2 Σµ , and of (locally) finite HN −2 measure. We therefore infer from the Radon-Nikodym theorem that Σµ = Θ∗ (.)HN −2

µ∗

Σµ ,

where θ∗ is the Radon-Nikodym derivative. In the same way, we obtain µ∗

(Ω \ Σµ ) = g(.)HN ,

and the proof is complete. Remark 2. Notice that working a little more the proof of Proposition 29, we could state that actually g(.) = |∇Φ|2 (.) for some harmonic function Φ defined on Ω.

4 Potential and modulus estimates The energy eε consists of two parts: the kinetic one, |∇uε |2 /2, and the potential one, Vε (uε ). The kinetic energy arises from variations in both the phase and the modulus of uε . In this section, we will prove that under assumption (H0 ) only the gradient of the phase contributes to µ∗ , the modulus and potential parts remaining uniformly bounded. More precisely, we will prove the following theorems (which are new in dimension N ≥ 3): Theorem 35 Let uε be a solution of (GL)ε on Ω such that assumption (H0 ) is verified. For any K ⊂ Ω compact, there exist a constant C(K, M0 ) depending only on K and M0 such that Vε (uε ) ≤ C(K, M0 ). K

and Theorem 36 Let uε be a solution of (GL)ε on Ω such that assumption (H0 ) is verified. For any K ⊂ Ω compact, there exist a constant C(K, M0 ) depending only on K and M0 such that 2 |∇|uε || ≤ C(K, M0 ). K

In the proofs of Theorems 35 and 36 we will need the following easy pointwise estimates, whose proofs are based on the maximum principle.


Lemma 3 Let uε be a solution of (GL)ε on Ω = B(R), R > a constant C such that |uε |(x) ≤ 1 + C Moreover, if |uε | ≥

1 2

309

√ ε. There exist

ε2 . dist(x, ∂Ω)

on B(R), then

|uε | ≥ 1 − C

ε2 − Cε2 |∇ϕε |2L∞ (B(R)) R2

on B(R/2), where uε = ρε exp(iϕε ). Proof of Theorem 35. Let K ⊂ Ω be given and δ = dist(K, ∂Ω). We decompose K in two parts K1 and K2 , where 1 K1 = x ∈ K s.t. dist(x, {|uε | ≤ 1 − σ}) ≤ ε 8 and K2 = K \ K1 . If x ∈ K1 , we infer from Proposition 25 that there exists 1 r(x) ∈ (ε 16 , δ/2) such that ˜ε (uε , x, δ/2) E C log Vε (uε ) ≤ eε (uε ). (34) 1 ˜ε (uε , x, ε 16 |log ε| ) E B(x,r(x)) B(x,r(x)) On the other hand, by monotonicity and assumption (H0 ) we have ˜ε (uε , x, δ/2) ≤ Cδ 2−N M0 |log ε|, E

(35)

and since x ∈ K1 , we also have 1 ˜ε (uε , x, ε 16 E ) ≥ Cη0 |log ε|.

(36)

Indeed, to obtain the last inequality observe that by assumption there exists 1 x ˜ ∈ {|uε | ≤ 1− σ} such that |x − x ˜| ≤ ε 8 . Therefore, applying the clearing-out theorem at the point x ˜, we obtain 1 16

˜ε (uε , x, ε ) ≥ E

2−N

1

ε 16 1

1

ε 16 − ε 8

1 1 η0 ˜ε (uε , x ˜, ε 16 − ε 8 ) ≥ |log ε| E 2

for ε sufficiently small. Combining (34),(35) and (36) we are led to C(M0 , K) Vε (uε ) ≤ eε (uε ). |log ε| B(x,r(x)) B(x,r(x))

(37)

From the open covering {B(x, r(x))}x∈K1 of K1 , we may extract a Besicovitch sub-covering, so that by addition we finally deduce that

310

Didier Smets

C(M0 , K) Vε (uε ) ≤ |log ε| K1

eε (uε ) ≤ C(K, M0 ).

(38)

Ω 1

We are left with the estimate on K2 . Let x ∈ K2 and set R = ε 8 . We deduce from Proposition 29 that |∇ϕε |2L∞ (B(x,R/2)) ≤ C(K)M0 ε− 4 |log ε|. 1

Hence, it follows from Lemma 3 that on B(x, R/4), 7

1 − C(K)(1 + M0 |log ε|)ε 4 ≤ |uε | ≤ 1 + C(K)ε2 . In particular Vε (uε ) ≤ C(K, M0 )ε

in K2 ,

so that the conclusion follows by integration.

Proof of Theorem 36. Let K ⊂ Ω be given and let χ be a cut-off function supported in Ω such that χ ≡ 1 on K. Define vε = χuε . Then, 1 uε (1 − |uε |2 )χ − uε ∆χ − 2∇uε · ∇χ ≡ fε . ε2 Note that in view of Theorem 35, −∆vε =

fε L2 (Ω) ≤ C(K, M0 )ε−1 . By elliptic regularity theory, we therefore deduce that vε H 2 (Ω) ≤ C(K, M0 )ε−1 . On the other hand, since |vε | ≤ C(K), by the Gagliardo-Nirenberg inequality 1

1

2 2 ∇vε L4 (Ω) ≤ vε H 2 (Ω) vε L∞ (Ω) ,

we infer from the previous estimate that ∇vε L4 (Ω) ≤ C(K, M0 )ε− 2 . 1

(39)

The modulus ρ ≡ |uε | verifies the equation −∆(ρ2 ) + 2|∇uε |2 =

1 2 ρ (1 − ρ2 ). ε2

(40)

Multiplying (40) by (1 − ρ2 )χ and integrating by parts we obtain |∇(ρ2 )|2 ≤ 2 (1 − ρ2 )χ2 |∇uε |2 K Ω 2 2 ≤ 2 (1 − ρ )|∇vε | + 2 (1 − ρ2 )|∇χ|2 |uε |2 Ω

≤ ε ∇vε 2L4

12

Vε (uε ) suppχ

≤ C(K, M0 ),

Ω

12

+ Cε

Vε (uε ) suppχ


311

where we have used Theorem 35 and (39). Finally, 1 1 |∇ρ|2 = |∇(ρ2 )|2 + (1 − ρ2 )|∇(ρ2 )|2 2 K 2 K K 12 12 4 ≤ C(K, M0 ) + Cε Vε (uε ) |∇uε | K

K

≤ C(K, M0 ), and the proof is complete.

5 Σµ is a minimal surface 5.1 Classical and weak notions of mean curvature Let Σ be a smooth compact manifold of dimension k, and γ0 : Σ → RN (N ≥ k) a smooth embedding, so that Σ 0 = γ0 (Σ) is a smooth k-dimensional submanifold of RN . The mean curvature vector at the point x of Σ 0 is the vector of the orthogonal space (Tx Σ 0 )⊥ given by ⎛ ⎞ N −k k N −k α ∂ν ⎝ (τj · HΣ 0 (x) = − )ν α ⎠ = − (divTx Σ 0 ν α ) ν α , (41) ∂τ j α=1 α=1 j=1 where (τ1 , ..., τk ) is an orthonormal moving frame on Tx Σ 0 , (ν 1 , ..., ν N −k ) is an orthonormal moving frame on (Tx Σ 0 )⊥ , and divTx Σ 0 denotes the tangential divergence at the point x. The integral formulation of (41) is given by k 0 divTx Σ X dH = − HΣ 0 · X dHk , (42) Σ0

Σ0

for all X ∈ Cc∞ (RN , RN ). The vectors HΣ 0 (·) are uniquely determined by (42), and in particular the definition in (41) does not depend on the choice of orthonormal frames. Assume now that Σ is only rectifiable. Then for Hk -a.e. x ∈ Σ, there exist a unique tangent space Tx Σ belonging to the Grassmanian GN,k . The distributional first variation of ν is the vector-valued distribution δν defined by divTx Σ Xdν for all X ∈ Cc∞ (RN , RN ). (43) δν(X) = Σ

In case |δν| is a measure absolutely continuous with respect to ν, we say that ν has a first variation and we may write δν = Hν, where H is the Radon-Nikodym derivative of δν with respect to ν. In this case, formula (43) becomes

312

Didier Smets

divTx Σ Xdν = −

Σ

H · X dν.

(44)

Σ

Remark 3. Notice that in the smooth case, this notion coincides with the definition (41), in view of (42). Notice also that the component of H which is orthogonal to Tx Σ is independent of the density Θ. However, if Θ is non constant, then H may have a tangential part. 5.2 Deriving the curvature equation Let X ∈ D(Ω, RN ) be a smooth vector field. We have eε (uε )divX = − ∇eε (uε ) · X Ω Ω 1 1 ∇(|∇uε |2 ) + 2 (1 − |uε |2 )(−2uε ∇uε ) · X, =− 2 2ε Ω

(45)

and ∂ 2 uε ∂uε ∂uε ∂ 2 uε i ∂uε ∂uε ∂X i =− ( + )X ∂xi ∂x2j Ω i,j ∂xi ∂xj ∂xj Ω i,j ∂xi ∂xj ∂xj 2 ∂ ∂uε i =− ∇uε · X∆uε − ∂xj X ∂x i Ω Ω i,j 1 ∇(|∇uε |2 ) · X. =− ∇uε · X∆uε − 2 Ω Ω

(46)

Since uε is a solution of (GL)ε , we deduce from (45) and (46) that 1 |log ε|

∂uε ∂uε ∂X i ∂xi ∂xj ∂xj 1 1 = (∇uε · X) ∆uε + 2 uε (1 − |uε |2 ) = 0. |log ε| Ω ε

eε (uε )δij − Ω

(47)

In view of the last formula, we will analyze the weak limit of the stressenergy tensor ∇uε ⊗ ∇uε αε = Id − dµε . eε (uε ) Clearly, |αε | ≤ CN µε , and we may assume that αε α∗ ≡ A · µ∗ , where A is an N ×N symmetric matrix. Since the symmetric matrix ∇uε ⊗∇uε is non-negative, we have A ≤ Id. (48)


313

On the other hand, T r(eε (uε ) Id − ∇uε ⊗ ∇uε ) = (N − 2)eε (uε ) + 2Vε (uε ). Therefore, since the trace is a linear operation, passing to the limit we obtain, using Theorem 35, T r(A) = (N − 2). (49) Passing to the limit in (47), and using the decomposition in Theorem 34 (see also Remark 2, we obtain |∇Φ∗ |2 ∂Φ∗ ∂Φ∗ ∂X i ∂X i δij − Aij d(µ∗ Σµ ) + dx = 0. (50) ∂xj 2 ∂xi ∂xj ∂xj Ω Ω On the other hand, Φ∗ is harmonic, −∆Φ∗ = 0.

(51)

Multiplying (51) by X · ∇Φ∗ , we obtain |∇Φ∗ |2 ∂Φ∗ ∂Φ∗ ∂X i δij − dx = 0. 2 ∂xi ∂xj ∂xj Ω

(52)

Combining (50) and (52) we have therefore proved Lemma 4 For every X ∈ Cc∞ (Ω, RN ), ∂X i Aij d(µt∗ ∂x j Ω

Σµ ) = 0.

(53)

Recall that we already know that Σµ is rectifiable. Comparing (53) with (44), it is tempting to prove that the matrix A corresponds to the orthogonal projection P onto the tangent space Tx Σµ . Lemma 5 We have 2

3

∇χ(y) dHN −2 (y) = 0

A(x)

for HN −2 -a.e. x ∈ Σµ ,

(54)

Tx Σµ

and for all χ ∈ Cc∞ (Ω, R). Proof. Let x ∈ Σµ be such that Tx Σµ exists and such that x is a Lebesgue point for Θ∗ (with respect to HN −2 ) and for A (with respect to µ∗ Σµ ). For r > 0, consider the vector field Xr,l (y) = χ( x−y r )el . Inserting Xr,l into (53) and letting r → 0, we obtain the desired result. A straightforward consequence is Corollary 37 For x as in Lemma 5, ⊥

(Tx Σµ ) ⊆ Ker A(x).

314

Didier Smets

With a little more elementary linear algebra, we further deduce Corollary 38 For x as in Lemma 5, A = P is the orthogonal projection onto the tangent space Tx Σµ . Proof. By (48), A ≤ Id, and therefore all the eigenvalues λ1 , ..., λN of A N are less or equal to 1. By (49), Tr(A) ≥ N − 2, so that i=1 λi ≥ N − 2. On the other hand, by Corollary 37, A has at least two eigenvalues, say λ1 and λ2 , equal to zero. Therefore, λi = 1 for i = 3, ..., N . In particular A is an orthogonal projection on an (N-2)-dimensional space. Since Ker A(x) ⊇ (Tx Σµ )⊥ , and since dim(Tx Σµ ) = N − 2, the conclusion follows. Combining the previous arguments, we have finally proved Theorem 39 The measure µ∗

Σµ has a first variation and

δ(µ∗

Σµ ) = 0,

i.e. (Σµ , Θ∗ ) is a stationary varifold.

References 1. L. Ambrosio and M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 25 (1997), 27-49. 2. F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices, Birkh¨ auser, Boston, 1994. 3. F. Bethuel, H. Brezis and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal. 186 (2001), 432-520. Erratum 188 (2002), 548-549. 4. F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS) 6 (2004), 17-94. 5. F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic GinzburgLandau equation to motion by mean curvature, Annals of Mathematics 163 (2006), 37-163. 6. Y. Chen and M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989), 83–103. 7. D. Preiss, Geometry of measures in Rn : distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537-643. 8. R. Schoen and S.T. Yau, Lectures on harmonic maps, International Press, Cambridge, MA, 1997. 9. L. Simon, Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Anal., Austr. Nat. Univ., 1983.

Index

Γ -convergence, 8, 177, 269 Γ -convergence of Ginzburg-Landau, 271 Γ -convergence along a trajectory, 277 Γ -convergence of gradient-flows, 275 Γ + -convergence, 245 tree-like Gasket, 152 approximate differentiability, 92 approximate discontinuity set, 91 approximate jump point, 91 approximate limit, 91 asymptotic expansion in Γ + convergence, 256 Bernoulli free boundary problem, 243 Blake-Zisserman model, 36 Bochner identity, 299 boundary layer, 22 Cantor part, 91 Cantor set, 152 capacitary potential, 239 capacity, 239 clearing out, 299 co-area formula, 90, 122 coerciveness, 82 collisions of vortices, 289 compact set, 82 Concentration-compactness alternative, 237 conservative solution, 278 convergence of discrete functions, 8 convex functional, 8

correctors, 215 crease energy, 52 curvature equation, 312 De Giorgi’s Rectifiability Theorem, 90 De Giorgi-Letta measure criterion, 121 density, 297 diffusion driving process, 225 direct methods, 83 Dirichlet form, 155 dynamics of vortices, 286 eigenform of a fractal, 161 finitely ramified fractals, 152 function of bounded variation, 58, 88 function, discrete, 8 fundamental solution, 239 Gasket, 142 generalized Sobolev inequality, 244 Ginzburg-Landau simplified model, 269 gradient-flow solution, 278 Green’s function, 253 harmonic center, 255 harmonic extension, 150 harmonic radius, 255 harmonic transplantation, 260 higher-order Γ -limit, 55 homogenization of perforated domains, 191 homogenization on the Gasket, 173 homogenization, discrete, 34

316

Index

integral representation, 125 interior boundary, 90 invariant measure, 226 isoperimetric inequality for the capacity, 240 jump set, 38, 91 Lennard-Jones potentials, 52 localization of the generalized Sobolev inequality, 249 long-range interactions, 10, 14 lower density, 297 lower semicontinuity, 82 lower semicontinuous envelope, 83 maximum correlation coefficient, 212 mean curvature, 311 minimal surface, 311 mixing coefficients, 211 modulus estimates, 308 monotonicity formula, 295 multi-phase limit, 28, 29 nearest-neighbour interactions, 9, 11 nesting property, 156 next-to-nearest neighbour interactions, 12 non-local limit, 31 one-dimensional BV functions, 92 one-dimensional relaxation, 93 oscillating Dirichlet boundary conditions, 190 oscillating Fourier boundary conditions, 193 perimeter, 90 Perron-Frobenius Theorem, 183 phase transition, 22 piecewise-Sobolev functions, 38 Pohozaev identity, 295

potential estimates, 308 quasiconvex function, 113 recovery sequence, 246 rectifiable set, 90, 307 reduced boundary, 90 regularity theorems for GinzburgLandau, 299 relaxation, 83 renormalization factor, 163 renormalization group, 58 renormalization operator, 164 renormalized energy, 273 rescaled energy, 295 Robin function, 254, 263 SBV, 85 scaling properties, 294 second-difference energies, 35 sequential coerciveness, 82 set of finite perimeter, 90 Sierpinski Carpet, 152 Sierpinski Gasket, 142 small energy regularity, 299 snowflake, 152 Sobolev inequality, 235 special functions of bounded variation, 85, 92 strange term, 190 strongly connected fractal, 181 subadditive envelope, 96, 132 uniform mixing coefficient, 212 upper density, 297 vector measure, 86 Vicsek set, 152 vortex, 268 weak mean curvature, 311 weak∗ convergence, 89

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